London is the most historically layered city in northern Europe and the most continuously inhabited: a Roman trading post founded in 43 CE, destroyed by Boudica in 60 CE, rebuilt, made capital of Roman Britain, settled by Saxons, conquered by Vikings, taken by William the Conqueror in 1066, burned in 1666, bombed in 1940–41, and rebuilt each time with a metropolitan confidence that has never been shaken. The city of 9 million people that exists today preserves this sequence of destructions and reconstructions in its street plan: the Roman walls, the medieval lanes of the City (the original square mile), the Georgian squares of Mayfair, the Victorian terraces of Notting Hill, the brutalist towers of the South Bank and the glass skyscrapers of Canary Wharf — all visible simultaneously from the viewing gallery of The Shard, and all contributing to the specific quality of London that makes it unlike any other European capital.

The British Museum — founded 1753, the first public national museum in the world, housing 8 million objects spanning 2 million years of human history — is simultaneously the most ambitious institution in the country and the most impossible to see completely: the Rosetta Stone, the Elgin Marbles, the Sutton Hoo helmet, the Lewis Chessmen, the Portland Vase, the Assyrian Lion Hunt reliefs from Nineveh, the Lindow Man — each a category-defining object in its field, arranged in a building whose Great Court (Foster + Partners, 2000, a glass-roofed enclosed courtyard that is the largest covered public square in Europe) is itself the finest modern intervention in any London institution. The National Gallery (1824, Trafalgar Square, free), the Tate Modern (2000, Bankside Power Station), the Victoria and Albert Museum (the world's greatest decorative arts collection, free) and the Natural History Museum (Alfred Waterhouse, 1881, with its extraordinary Romanesque terracotta façade, free) together constitute the most concentrated free museum offer in any capital city in the world.

98types Studio is in Camden Market (Market Hall, Camden Lock Place, NW1 8AL) — the specific neighbourhood where the city's alternative cultural identity is most permanently expressed: the Lock Market's Victorian canal infrastructure, the Horse Tunnel's tattoo parlours and vinyl shops, the Electric Ballroom's concert history, the market's 1,000 independent stalls selling everything from Japanese street food to hand-dyed silk. The studio has been here for 14 years. The London city street map print from 98types captures the full grid of the capital — from the City of London and Canary Wharf in the east to Notting Hill and Kensington in the west, from Hampstead Heath in the north to Greenwich in the south. It is the largest and most detailed UK map in the collection, and the most frequently bought as a gift.

Edinburgh is the most dramatically situated capital in Europe — a city built around three volcanic ridges, its medieval Old Town running along the spine of the Royal Mile from the Castle to the Palace of Holyroodhouse, its Georgian New Town laid out on the adjacent ridge from 1767 in the most complete example of Enlightenment urban planning in the world. The tension between these two cities — the dark, organic medieval city with its closes and wynds and its history of plague, reformation and public execution, and the rational, classical New Town with its ordered squares and elegant Georgian terraces — gives Edinburgh its specific intellectual character: a city that has always been simultaneously ancient and progressive, simultaneously Catholic and Presbyterian, simultaneously Scottish and British.

The Edinburgh Castle (on its 135-metre volcanic plug, inhabited since the 12th century, currently housing the Scottish Crown Jewels, the Stone of Destiny and the National War Museum of Scotland) is the starting point of the Royal Mile — a kilometre of medieval high street connecting the Castle to the Palace of Holyroodhouse (the official Scottish residence of the British monarch), passing through the Lawnmarket (the former market for linen from the Lothians), the High Street (St Giles' Cathedral, the Heart of Midlothian cobblestone heart where medieval prisoners were executed, the Camera Obscura) and the Canongate (John Knox's house, the Museum of Edinburgh, the Scottish Parliament building). The Royal Mile is the most compressed historical sequence in any UK city — a kilometre containing a thousand years of Scottish history.

The Edinburgh International Festival and its unofficial twin, the Edinburgh Festival Fringe (August, three weeks, 3,500 shows, 50,000 performances, 300 venues — the largest arts festival in the world by any measure), transform the city into the most intensely cultural urban experience available anywhere on earth: every pub, church, cellar, park and car park becomes a venue; the streets fill with performers; the city's normal population of 500,000 doubles. The whisky experience — the Scotch Whisky Experience on the Royal Mile, the Holyrood Distillery, the day trips to Speyside and Islay that begin from Edinburgh's Waverley Station — is equally serious and considerably less crowded.

Manchester is the city that made the modern world. Not metaphorically — the specific combination of the Industrial Revolution (the first steam-powered cotton mills in Ancoats, 1780s), the labour movement (the Peterloo Massacre of 1819, when cavalry charged a crowd demanding parliamentary reform in St Peter's Field — the site now occupied by the Midland Hotel — killing 18 people and injuring 700; the direct ancestor of trade unionism), the Free Trade movement (the Anti-Corn Law League, founded in Manchester 1838, which established the principle of free trade that shaped global economics for two centuries), and the women's suffrage movement (Emmeline Pankhurst, born Manchester, founded the WSPU in Manchester in 1903) make Manchester the city where the political and economic frameworks of modern liberal democracy were invented and contested.

The Manchester Museum of Science and Industry (the world's first passenger railway station, Liverpool Road, 1830 — the terminus of the Liverpool and Manchester Railway) is the most historically significant building in industrial Manchester, and the museum that occupies it is the most complete account of the Industrial Revolution available in any single institution. The Manchester Art Gallery (free, on Mosley Street) contains the finest collection of Pre-Raphaelite paintings outside London — including Ford Madox Brown's Work (1852–65) and Holman Hunt's The Hireling Shepherd (1851) — and the Whitworth Art Gallery (in Whitworth Park, a Waterhouse building from 1889 extensively renovated by MUMA in 2015) is among the finest regional galleries in the country.

Manchester's music history is the most concentrated in Britain: the Haçienda (the Factory Records nightclub on Whitworth Street West, opened 1982, where Madchester was invented, closed 1997, now apartments — a blue plaque on the wall is the only physical evidence); the Free Trade Hall (now a hotel, where Bob Dylan was called Judas in 1966 for going electric, and where the Sex Pistols played to an audience of 40 people in 1976 who subsequently formed most of the significant Manchester bands of the following decade); the Afflecks independent market in the Northern Quarter (where Ian Curtis, Morrissey, Mark E Smith and countless successors came to buy their records and clothes). The specific sound of Manchester — Joy Division, The Fall, The Smiths, Oasis, The Chemical Brothers — is the most influential regional music tradition in British pop history.

Bath is the only British city to have been declared a UNESCO World Heritage Site in its entirety — every building within the city boundary, every street, every garden, every view, protected as a complete ensemble of outstanding universal value. The specific quality that earned this designation is the city's extraordinary architectural coherence: the decision in the 18th century to build Bath almost entirely in the local honey-coloured Bath stone (a Jurassic oolitic limestone, quarried from Combe Down and Bathampton Down, warm gold in sunlight, silver-grey in rain) gave the city a visual unity that no other British city possesses. When the light is right — late afternoon in October, or morning in spring — the entire city glows as if internally lit.

The Roman Baths (Aquae Sulis, 1st century CE) are the best-preserved Roman thermal baths complex in the world north of the Alps: the Great Bath (a 1st-century lead-lined pool still fed by the same geothermal spring, at 45°C, that the Romans exploited), the sacred spring whose silt preserved 12,000 coins and 130 lead curse tablets thrown by worshippers of Sulis Minerva, the hypocaust (underfloor heating system), the frigidarium and the museum of finds. The Baths are immediately beneath the 18th-century Abbey Church, and the combination of Roman engineering and Gothic Christian architecture on the same site — both drawing meaning from the same hot spring — is the most compressed stratigraphic encounter between classical and medieval Britain available anywhere.

The Royal Crescent (John Wood the Younger, 1767–74) — a 500-metre arc of 30 identical town houses facing a sweeping lawn, the most theatrical piece of urban design in Britain — and The Circus (John Wood the Elder, 1754–68, a circular colonnade of three concentric streets facing inward, the plan possibly derived from Stonehenge) are the twin masterpieces of Georgian Bath: buildings designed not merely to house people but to make a philosophical statement about the relationship between classical order and natural landscape. The Pulteney Bridge (Robert Adam, 1774, one of only four bridges in the world with shops on both sides, based on the Ponte Vecchio in Florence) and the Thermae Bath Spa (Grimshaw Architects, 2006, a glass and Bath stone complex where the geothermal springs now power an open-air rooftop pool) complete a city where the Roman, the Georgian and the contemporary coexist with an ease that is uniquely Bath's own.

Cambridge is one of the most beautiful small cities in the world and the most densely concentrated repository of scientific and intellectual achievement in the history of Western civilisation. The University of Cambridge was founded in 1209 by scholars expelled from Oxford, and the 31 colleges that have accumulated on the river Cam since then constitute the most remarkable architectural and intellectual ensemble in Britain: Isaac Newton at Trinity (1661–67, where he worked out the laws of gravitation while watching an apple fall in the Fellows' Garden); Charles Darwin at Christ's (1828–31, where he began the observations that would become On the Origin of Species); Francis Crick and James Watson at the Eagle pub on Benet Street (1953, where they announced the double helix structure of DNA over their lunch of steak and kidney pudding).

The King's College Chapel (begun 1446, Henry VI's personal project, the perpendicular Gothic vault 88 feet above the nave the most technically ambitious stone ceiling ever built, the Rubens altarpiece below it the most valuable painting in any UK university) is the defining building of Cambridge and among the finest buildings in Britain. The Backs — the gardens behind the colleges that back onto the River Cam — constitute the most continuously beautiful sequence of garden and architecture in any British city: the Mathematical Bridge at Queens' (1749, designed to be self-supporting without bolts — a myth, but a persistent one), the Bridge of Sighs at St John's (1831, modelled on the Venetian original), the Clare Bridge (1640, the oldest surviving bridge in Cambridge) and the Wren Library at Trinity (Christopher Wren, 1695, containing Newton's notebook from his anni mirabiles and A.A. Milne's manuscript of Winnie-the-Pooh) all visible from the river in one uninterrupted sequence.

The Fitzwilliam Museum (George Basevi, 1848, free) is the finest regional art and antiquities museum in Britain: Egyptian sarcophagi, Greek painted pottery, Flemish Old Masters, Impressionists, illuminated manuscripts, and the William Blake collection — all in a Neoclassical building whose entrance hall is among the most gloriously excessive Victorian public interiors in the country. The Cambridge market square — the oldest continuously operating market in Britain, established in the 12th century, still operating six days a week — is the best single address for understanding what Cambridge is when it is not being a university: a working market town of 130,000 people whose economy is built on education, science (the Cambridge Biomedical Campus, the largest in Europe) and the most productive concentration of technology start-ups outside Silicon Valley.

Liverpool is the most emotionally engaged city in Britain and the most culturally overachieving relative to its size. A port city of 500,000 people on the Mersey estuary that dominated world trade from the 17th to the 20th century — the slave trade (the moral stain at the centre of the city's fortune, comprehensively documented in the International Slavery Museum at Albert Dock), then cotton, then general cargo — and that produced in a single decade (1960–1970) the most influential pop music in the history of recorded music: The Beatles, whose impact on global culture from John Lennon's bedroom in Menlove Avenue to the Ed Sullivan Show in 1964 to the White Album in 1968 cannot be meaningfully calculated.

The Albert Dock (Jesse Hartley and Philip Hardwick, 1846, the first non-combustible warehouse system in the world — cast iron columns, brick vaulting — now a UNESCO World Heritage site as part of Liverpool's Maritime Mercantile City) contains, in its five acres of converted warehouses, the Tate Liverpool (the largest display of modern and contemporary art outside London), the Merseyside Maritime Museum (the most comprehensive account of Liverpool's oceanic history, including the Titanic exhibition and the slavery galleries), and the Beatles Story museum. The Three Graces — the Royal Liver Building (1911, with its two Liver Birds atop the clock towers), the Cunard Building (1917) and the Port of Liverpool Building (1907) — form the most recognisable waterfront ensemble in Britain, and the view from the Mersey Ferry (still operating, as in the song) looking back at the Three Graces against a winter sky is the city's defining image.

The Liverpool sound — the specific musical culture that produced not just The Beatles but Gerry and the Pacemakers, Billy J. Kramer, Cilla Black, Echo and the Bunnymen, The Teardrop Explodes, Frankie Goes to Hollywood, The La's, Cast — is the most concentrated regional pop tradition in Britain and the most globally exported. The Cavern Club (Mathew Street, rebuilt on the original site after demolition for a ventilation shaft in 1973) is the most visited music venue in the UK — not for its present programme but for the 292 Beatles performances between 1961 and 1963 that created the mythology. The two Liverpool cathedrals — the Anglican (Giles Gilbert Scott, 1904–78, the largest cathedral in Britain, its tower visible 30km away) and the Metropolitan (Frederick Gibberd, 1967, nicknamed Paddy's Wigwam, a circular concrete crown-of-thorns design beloved by almost no one for 40 years and now considered a masterpiece) — face each other down Hope Street in the most architecturally charged 600 metres in any British city.