Paris is the city that has set the standard by which all other cities are judged for so long that the competition has become philosophical: not whether Paris is the best city in Europe, but what exactly it means for a city to be the best. The specific quality that Paris maintains, decade after decade, is a kind of formal seriousness about pleasure — the conviction, embedded in the architecture, the food, the museums, the parks, the manner in which streets meet squares meet rivers, that the quality of daily life is worth taking absolutely seriously and that beauty is not decorative but essential.

The Louvre is the largest art museum on earth and should be treated accordingly: not as a single visit but as a series of appointments. The Richelieu Wing for Dutch and Flemish painting; the Denon Wing for Italian Renaissance (the Winged Victory of Samothrace at the top of the staircase, then the Grande Galerie with its Leonardos, then the Mona Lisa — smaller than you expect, more interesting than the crowd around it); the Sully Wing for ancient Greek and Egyptian art. The Musée d'Orsay for Impressionism, housed in a converted railway station whose architecture is as extraordinary as its contents. The Centre Pompidou for modern and contemporary art, and for the view from its terrace.

Paris is also the city that has the best concentrated food culture of any capital in Europe: the specific combination of boulangeries, bistros, brasseries, caves à vin and épiceries fines that constitutes the Parisian neighbourhood food ecosystem is the template that every food city in the world has attempted to replicate and none has yet equalled. The Rue de Bretagne market in Le Marais, the Marché d'Aligre in the 12th, the cafés of Saint-Germain at seven in the evening with the lights coming on across the boulevard — these are the pleasures that make Paris worth the Eurostar fare every time.

Amsterdam is a city built on 11 million wooden piles driven into soft ground beneath the IJ bay — a northern Venice of canals, canal houses and bridges that managed to be both the most prosperous trading city in the world in the 17th century and the most architecturally coherent one in the 21st. The specific achievement of Amsterdam's urban form is the canal ring (grachtengordel): four concentric semicircular canals — Singel, Herengracht, Keizersgracht, Prinsengracht — dug between 1613 and 1662, lined on both sides with merchant houses whose narrow facades (taxed by width), stepped or bell gable tops, and iron lifting hooks represent the highest expression of Dutch Golden Age domestic architecture. The whole system is a UNESCO World Heritage site and looks, in every season and every light, like a Pieter de Hooch painting made civic.

The Rijksmuseum houses the national collection of Dutch Golden Age painting — Rembrandt's Night Watch (larger and darker than the reproductions suggest, the black background luminous rather than simply dark), Vermeer's The Milkmaid (the most quietly perfect painting of the 17th century), Jan Steen's The Merry Family. The Van Gogh Museum, a short walk away, contains the largest collection of Van Gogh's work in the world: 200 paintings, 500 drawings, in chronological rooms that track the development of one of the most extraordinary artistic careers in the history of painting from the dark Brabant interiors of the early 1880s to the blazing Provençal light of the Saint-Rémy period.

The Jordaan neighbourhood — west of Prinsengracht, a grid of tiny streets and smaller canals developed in the 17th century as a working-class district and now the most pleasant neighbourhood to walk in Amsterdam — contains the Anne Frank House, the concealed annexe where Anne Frank and her family hid for two years before their deportation, preserved exactly as it was in 1944. It is one of the most affecting places in Europe and should be booked weeks in advance.

Berlin is the European city that most explicitly bears the marks of its 20th-century history on its body. The city was divided for 28 years by a wall that ran through its centre, and the scars of that division — the empty spaces where buildings were demolished in the death strip, the deliberately preserved stretch of the Wall at the East Side Gallery, the unhealed join between two urban systems that developed in completely different directions — are visible to anyone who knows to look. This is not a disadvantage. It makes Berlin the most intellectually interesting capital in Europe to walk through, and the most honest about what the 20th century actually involved.

Museum Island — five major museums on a Spree island in the centre of Berlin, a UNESCO World Heritage site — is the most concentrated collection of world antiquities in Northern Europe. The Pergamon Museum houses the Pergamon Altar (180 BCE, from what is now Turkey), the Ishtar Gate of Babylon (575 BCE, reconstructed) and the Market Gate of Miletus (1st century CE) — complete ancient architectural monuments transported to Berlin in the 19th century and reassembled inside a museum purpose-built to contain them. The scale is staggering: you are not looking at ancient artefacts but walking through ancient architecture.

The Berlin Philharmonie — Hans Scharoun's 1963 concert hall, the most radical concert hall design of the 20th century — places the orchestra at the centre of a vineyard-terraced auditorium so that every seat in the house is within 30 metres of the conductor. The Berlin Philharmonic Orchestra is one of the finest in the world. A ticket is obtainable, the acoustics are extraordinary, and the combination of the architecture and the music is an experience that represents what Berlin does better than anywhere else: high culture that has broken free from convention without losing its seriousness.

Lisbon is the European capital that most rewards unstructured walking: a city of seven hills linked by staircases, funiculars, old cable cars and the specific urban texture of azulejo-tiled facades that make every street a mosaic composition. The specific quality of Lisbon light — Atlantic ocean light, softer and more diffused than Mediterranean light, with a golden late-afternoon quality that makes the blue-and-white tiles glow — is as much a feature of the city as any of its monuments, and the best time to experience it is in October, when the summer crowds have gone and the light reaches its most beautiful quality.

The Alfama — Lisbon's oldest neighbourhood, a Moorish-era labyrinth of narrow lanes and staircases climbing the hill above the Tagus — is where fado, the Portuguese musical tradition of melancholy longing (saudade), is still performed in small restaurants to audiences of local families and educated visitors. The Mouraria, adjacent, is the most multicultural neighbourhood in Lisbon, where African and South Asian communities mix with the traditional Portuguese working class in the city's most authentic and least tourist-adjusted neighbourhood. The Miradouro das Portas do Sol — one of the seventeen official viewpoints (miradouros) across the city — gives the most complete view of the Alfama, the Tagus and the hills opposite.

The Torre de Belém and the Mosteiro dos Jerónimos, both built at the height of the Age of Discovery in the early 16th century in the Manueline style (Portuguese Late Gothic, with its specific marine ornament of ropes, coral and armillary spheres), are the most important architectural monuments in Portugal. The Jerónimos monastery in particular — the cloisters are the finest Gothic cloister in the Iberian peninsula — should be seen in the morning light before the tour buses arrive.

Prague was never bombed in World War II — unlike Warsaw, Dresden, Coventry or Berlin — and the consequence of this accident of history is that it contains the most complete medieval and Baroque city centre in Central Europe, essentially unchanged since the 18th century. The Old Town (Staré Město), the Lesser Town (Malá Strana), the Hradčany castle district and the Jewish Quarter (Josefov) together constitute a UNESCO World Heritage site of almost absurd architectural richness: Gothic churches, Baroque palaces, Art Nouveau apartment buildings, Renaissance arcades, all within walking distance of each other, all intact, all still in daily use.

The Prague Castle complex — the largest ancient castle in the world by area — contains a Gothic cathedral (St Vitus, 600 years in construction), a Romanesque basilica, a Renaissance palace, Golden Lane (a row of tiny houses built into the castle walls in the 16th century), and the Story of Prague Castle museum, which documents the complete architectural and political history of the site. The view from the castle terrace over the red-tile rooftops of Malá Strana, across the Vltava to the spires of the Old Town, is the canonical view of Prague and still worth the walk up the hill.

Prague's classical music culture is among the richest in Europe: the Prague Spring Festival (May) is one of the major classical music festivals on the continent; the city's concert halls — the Art Nouveau Municipal House, the Rudolfinum, the historic Estates Theatre (where Mozart conducted the premiere of Don Giovanni in 1787) — offer performances at prices that are a fraction of what the equivalent concert would cost in London or Paris. Czech beer culture — the Pilsner tradition was invented in Plzeň, 90 minutes from Prague — is the world standard for lager, and the combination of excellent live music, outstanding beer and one of the most beautiful cities in Europe makes Prague one of the highest-value city breaks available from the UK.

Vienna was, for six centuries, the capital of the Habsburg Empire — which at its peak encompassed much of Europe from Spain to Transylvania — and the cultural ambition of that empire still manifests in the extraordinary density of world-class institutions concentrated in a compact city centre. The specific quality that distinguishes Vienna from other European capitals is the combination of imperial scale and domestic elegance: the Ringstrasse (the grand ceremonial boulevard built by Franz Joseph in the 1860s) lined with monumental institutions, and behind it, the Innere Stadt (first district) of narrow streets, coffeehouses and Baroque churches that represents daily Viennese life at its most characteristic.

The Kunsthistorisches Museum — the Art History Museum, built to house the Habsburg collections in 1891 — contains the most important Bruegel collection in the world (twelve of the surviving 45 works, in a single room), the finest collection of 16th-century German painting outside Germany, and the Egyptian and Near Eastern antiquities. The Vienna Secession building (1897) houses Klimt's Beethoven Frieze — the most complete Symbolist artwork in Austria — in its basement. The Belvedere Palace contains Klimt's The Kiss (the most reproduced painting in Austrian art), a collection of Schiele, and one of the most beautiful garden complexes in Central Europe.

Vienna's Kaffeehauskultur — the specific tradition of the Viennese coffeehouse as a place of intellectual work, newspaper reading, chess playing and extended social encounter — is a UNESCO Intangible Cultural Heritage. The historic coffeehouses — Café Central (where Trotsky, Freud and Stefan Zweig were regulars), Café Hawelka (unchanged since 1939), Café Landtmann (the most formal, opposite the Burgtheater) — serve melange (espresso with steamed milk), apple strudel and the specific atmosphere of a city that has always taken the pleasures of sitting down seriously.

Budapest is two cities unified in 1873: Buda, the hilly right bank of the Danube with its medieval castle district, and Pest, the flat left bank where the commercial city developed in the 19th century at extraordinary speed, producing a complete inventory of Neo-Gothic, Neo-Baroque, Art Nouveau and Eclectic architecture that rivals Vienna in ambition and exceeds it in flamboyance. The Hungarian Parliament Building — designed by Imre Steindl, completed 1904, the largest parliament building in Europe — is a Neo-Gothic fantasy of 691 rooms, 19km of stairways and 40kg of gold, sitting on the Pest bank of the Danube with a symmetry and scale that makes it simultaneously absurd and magnificent. From the Buda bank at night, with the parliament and the Chain Bridge lit and the Danube reflecting everything, it is one of the most extraordinary urban panoramas in Europe.

The thermal bath culture of Budapest is the most developed in Europe — the city sits on 80 thermal springs and has been exploiting them since the Roman occupation. The Széchenyi (outdoor and indoor, Art Nouveau, in the Városliget park), the Gellért (indoors, Art Nouveau, attached to a grand hotel, with the most ornate pool in the city) and the Rudas (originally 16th-century Ottoman, with a central octagonal hot pool under a dome) represent three completely different approaches to what a thermal bath should be. The Rudas is open mixed-gender on weekends and has a rooftop pool overlooking the Danube.

The ruin bars of the Jewish Quarter in Pest — Szimpla Kert being the most famous, a decaying courtyarded apartment block that became an alternative bar in 2002 and has been a cultural institution since — are Budapest's specific contribution to European bar culture: large, labyrinthine, deliberately unrestored, with mismatched furniture, outdoor spaces, live music and art installations coexisting in a way that feels accidental but is actually carefully curated.

Dubrovnik is the most complete surviving medieval fortified city on the Adriatic — a walled city-state that maintained its independence as the Republic of Ragusa for five centuries through a combination of diplomatic skill, merchant acuity and the most sophisticated city administration in the medieval Mediterranean (Ragusa abolished slavery in 1416, almost four centuries before Britain; introduced quarantine regulations in 1377; and established a health service, old people's home and pharmacy — the oldest still operating in Europe, the Rupe Pharmacy, founded 1317 — that would not be matched by any other European city for centuries). The walls that surround the old city, begun in the 8th century and completed in the 16th, are the most perfectly preserved medieval city walls in the world.

Walking the Stradun — the main limestone-paved street of the old city, gleaming white in sunlight, lined with Baroque buildings rebuilt after the earthquake of 1667 — is the most immediate experience of what a medieval city without cars feels like. The limestone absorbs and reflects light in a specific way; the scale of the street is exactly human; the buildings on either side were designed as an ensemble rather than as individual statements. At each end of the Stradun: the Pile Gate (the city's main entrance, drawbridge still intact) and the Ploca Gate (the harbour gate, leading down to the old port where fishing boats and island ferries dock).

The islands visible from the city walls — Lokrum (15 minutes by boat, uninhabited, with a salt lake and peacocks and the ruins of a medieval monastery), Kolocep, Lopud and Sipan (collectively the Elaphiti Islands, 30-45 minutes by boat, quiet villages, restaurants serving grilled fish on terraces over the sea) — provide the counterpoint to the intensely historical city. Dubrovnik without a boat trip to at least one island is the wrong Dubrovnik.

Reykjavík is the northernmost capital city in the world, a city of 230,000 people on a volcanic peninsula jutting into the North Atlantic, where the geological forces that created Iceland are still visibly at work: the steam vents and lava fields visible from the ring road, the geysers an hour's drive from the city centre, the glaciers covering 11% of the island's surface. Iceland sits directly on the Mid-Atlantic Ridge — the boundary between the North American and Eurasian tectonic plates — and is growing by 2cm per year as the plates spread apart. Walking between the tectonic plates at Þingvellir (30 minutes from Reykjavík) is the most geologically vertiginous experience available on a tourist day trip anywhere on earth.

The city itself — colourful corrugated-iron houses, the Hallgrímskirkja church (designed to evoke the basalt columns of Iceland's lava fields, visible from every point in the city), the harbour with its whale-watching boats and puffin-spotting tours, the Harpa concert hall (Henning Larsen Architects, 2011, a glass facade of hexagonal cells that changes colour with the light across the day) — is a small city in which the relationship between human habitation and extraordinary natural environment is more explicitly stated than anywhere else in Europe. The sky over Reykjavík is the sky of the North Atlantic, vast and variable, and when the Northern Lights appear over the harbour in winter, green and purple curtains shifting against a sky of hard stars, the city achieves a quality of beauty that has no equivalent in European urban experience.

The Golden Circle — Þingvellir national park (site of the world's oldest parliament and the tectonic rift), the Geysir geothermal area (where the Great Geysir gave its name to every geyser on earth and its neighbour Strokkur erupts every 5-10 minutes to 30 metres) and the Gullfoss waterfall (the Hvítá river dropping 32 metres into a gorge of absolute fury) — is the essential Iceland day trip and one of the most geologically extraordinary landscapes accessible by tourist vehicle on the planet.

Athens is the city where Western civilisation began and where the physical evidence of that beginning is still standing, visible from across the city, on a limestone hill that has been continuously occupied since the Neolithic period. The Acropolis of Athens — with the Parthenon (447–432 BCE), the Erechtheion (421–406 BCE), the Temple of Athena Nike (424 BCE) and the Propylaea (437–432 BCE) — is the most significant surviving architectural ensemble of ancient Greece and one of the most significant in the history of human culture. The Parthenon is not merely a beautiful old building: it is the source of the proportional system, the visual language and the philosophical argument about the relationship between mathematics and beauty that has informed Western architecture for 2,500 years.

The Acropolis Museum (2009, Bernard Tschumi Architects), built over an archaeological excavation that is visible through its glass floors, houses the surviving sculptures from the Acropolis in climate-controlled conditions. The top floor — a 1:1 recreation of the Parthenon's exterior at the exact orientation of the original, with the Acropolis visible through the glass walls — displays the surviving frieze panels alongside plaster casts of the panels held in the British Museum, the Louvre and elsewhere. The question of where the Elgin Marbles belong is answered visually, without argument, by the gap between the originals and the casts in the top-floor gallery.

The Monastiraki neighbourhood at the foot of the Acropolis hill is the best place in Athens to eat: the flea market, the tavernas serving meze on tables that spill onto the square, the souvlaki grills that have been operating since before the streets were paved. The Psyrri neighbourhood adjacent is where Athenian nightlife concentrates: small restaurants in converted warehouses, bars in neoclassical buildings, the specific energy of a southern European city that treats eating and drinking as the serious business they are.