Marrakesh is the city where the medieval world and the present occupy the same street simultaneously, without apparent discomfort. The medina — the old walled city, a UNESCO World Heritage site since 1985 — was built in the 11th century and has been continuously inhabited since, its street pattern the product of centuries of organic development that has produced the most complex urban labyrinth in North Africa: lanes that narrow to a shoulder's width, open into small squares with fountains, dead-end into dyers' workshops or leather tanneries, and then, without warning, emerge into the Djemaa el-Fna — the great central square that is simultaneously the largest open-air theatre, the largest food market and the largest gathering of itinerant performers in the Islamic world.

The Djemaa el-Fna has been the social centre of Marrakesh for a thousand years and it is, in its present form, a UNESCO Intangible Cultural Heritage: the hlaykiyya storytellers (telling tales in Darija Arabic to crowds of locals, often for hours, incomprehensible to tourists but compelling in their performance), the gnawa musicians (a specific Moroccan tradition with West African roots, the trance music of the sub-Saharan diaspora), the snake charmers, the henna artists, the orange juice vendors with their precisely stacked pyramids of Moroccan oranges. At night the square transforms: 100 food stalls erect themselves in a choreography that takes less than an hour, and the smoke and the calls and the light turn the Djemaa into something between a restaurant and a medieval festival.

Beyond the square: the Saadian Tombs (1578, closed by the succeeding dynasty and sealed for 200 years, rediscovered and reopened in 1917 — the most lavish funerary architecture in Morocco), the Medersa Ben Youssef (the largest Islamic school in the Maghreb, founded in the 14th century, its courtyard of carved cedar, stucco and tile among the finest interiors in Morocco), and the Palais de la Bahia (a 19th-century vizier's palace of 160 rooms, seven courtyards, acres of painted cedar and zellij tilework — the most theatrical domestic architecture in Africa). The souks — the specialist markets of the medina, each street devoted to a single trade, the coppersmiths, the dyers, the weavers, the leatherworkers — are the best place in the world to understand how a medieval city actually worked, because they still do.

Cairo is the most historically layered city on earth — a place where 5,000 years of continuous civilisation have been deposited in geological strata that are simultaneously visible, accessible and still in daily use. The pyramids of Giza are 30 minutes from the city centre by taxi. The Islamic old city — a UNESCO World Heritage site of unparalleled density, with 600 listed monuments from the 7th to the 19th centuries within a single neighbourhood — is 20 minutes from the pyramids. The Coptic quarter of Old Cairo, with its 3rd-century churches built into a Roman fortress, is 10 minutes from the Islamic city. Cairo is not a museum: it is a capital of 21 million people where the monuments are embedded in a living urban fabric and where the distinction between past and present is the least legible of any city in the world.

The Pyramids of Giza require no introduction but reward careful preparation: the Great Pyramid of Khufu (the only surviving Wonder of the Ancient World, built around 2560 BCE, containing 2.3 million blocks of limestone and granite at an average weight of 2.5 tonnes each) is the single most studied building in human history and still the most impenetrable in terms of how it was actually built. The Sphinx (2558 BCE, carved from a single limestone outcrop) has been buried up to its neck in sand for most of its existence and reveals its full 73-metre length and 20-metre height only in recent centuries. The view from the Solar Boat Museum terrace at sunrise, looking east from the plateau over the entire city of Cairo in the morning haze, is the most spatially overwhelming experience available in any African city.

The Egyptian Museum in Tahrir Square (1902, now supplemented by the new Grand Egyptian Museum at Giza, 2023, the largest archaeological museum in the world) houses the greatest collection of ancient Egyptian artefacts anywhere: the treasures of Tutankhamun (the golden death mask, the gilded funerary shrine, the 5,398 objects found in his tomb in 1922), the Royal Mummies Room (18 pharaohs in glass cases, including Ramesses II, the probable pharaoh of the Exodus), and the Narmer Palette (3100 BCE, the oldest historical document in existence). The Khan el-Khalili bazaar in the heart of Islamic Cairo — functioning continuously since 1382, specialising in spices, perfumes, silver and textiles — is the oldest commercial institution in the Arab world and still the most atmospheric market in Cairo.

Cape Town is the most naturally beautiful city in the world — a statement that is genuinely contestable only by Sydney, San Francisco and Rio de Janeiro, and that most people who have been to all four tend to resolve in Cape Town's favour. The specific quality of the city's setting — the flat-topped basalt plateau of Table Mountain (1,086 metres, its summit frequently draped in the "tablecloth" of cloud that forms as southerly winds hit the mountain) rising directly from the city centre to the south, the Atlantic to the west, the Indian Ocean to the east, the Cape of Good Hope headland 50km to the south — gives it a geographical drama that no human intervention could have achieved.

The city's human history is as complex as its landscape is beautiful: Cape Town was the point of European entry into sub-Saharan Africa (Jan van Riebeeck, Dutch East India Company, 1652), the administrative capital of the Cape Colony, the hub of the slave trade in southern Africa, and the site of Robben Island — the maximum-security prison where Nelson Mandela spent 18 of his 27 years of imprisonment, visible from the V&A Waterfront, now a UNESCO World Heritage site and museum accessible by ferry. The District Six Museum in the city centre documents the forced removals of the apartheid era — the 60,000 residents of a mixed-race inner city community who were evicted from 1968 onwards — with a specific intimacy and moral seriousness that makes it one of the most important human rights museums in the world.

The Cape Winelands — Stellenbosch (45 minutes from the city), Franschhoek (75 minutes), Paarl and the Hemel-en-Aarde Valley near Hermanus — constitute the most scenically dramatic wine-producing region in the world: vineyards in mountain valleys with the specific light of the Cape, producing Pinotage, Chenin Blanc, Syrah and Cabernet Sauvignon of genuine international quality. The combination of a morning on Table Mountain, an afternoon in Stellenbosch and an evening of Cape Malay cuisine (the specific fusion of Indonesian, Indian and African cooking brought by enslaved workers from the East) in the Bo-Kaap neighbourhood is the complete Cape Town day — culture, nature, history and food simultaneously.

Johannesburg is the city that built modern South Africa and the city that best explains its contradictions. Founded in 1886 on the discovery of the Witwatersrand gold reef — the largest gold deposit ever found — it grew from a mining camp of 3,000 people to a city of 5 million in 130 years, accumulating in that time the entire history of South African apartheid and the specific aftermath of liberation: a city of extraordinary wealth and extraordinary poverty in the same postcodes, of world-class art institutions and still-dangerous townships, of the most important museum of 20th-century political history in Africa and the suburb where Nelson Mandela chose to live after his release from prison.

The Apartheid Museum (2001, near Gold Reef City, Ormonde) is the most important museum in South Africa and one of the most important in the world: a comprehensive documentation of the apartheid system (1948–1994) using archive photographs, film footage, testimonies and physical artefacts arranged in a sequence that moves from the legislative foundations of racial classification through the Sharpeville Massacre, the Soweto Uprising and the resistance movements to the negotiated transition and the miracle of 1994. The museum was designed to be experienced sequentially, and each room builds on the last with a dramatic intelligence that ranks it among the finest museum design achievements of the 21st century. Entry is through two doors: one marked WHITES, one marked NON-WHITES. Your ticket tells you which one to use.

Soweto — the South Western Township, 15km from the city centre — is simultaneously a post-apartheid city of 1.3 million people, a living neighbourhood of extraordinary vitality, and the most historically significant urban area in South Africa: the site of the 1976 Soweto Uprising (when students protesting compulsory instruction in Afrikaans were fired on by police, beginning the chain of events that eventually ended apartheid), the location of the Hector Pieterson Museum and the street where both Nelson Mandela and Archbishop Desmond Tutu had their homes on the same block (the only street in the world to have housed two Nobel Peace Prize winners simultaneously). The Cradle of Humankind — 50km northwest of the city, a UNESCO World Heritage site containing fossilised hominid remains from 3.5 million years ago, including Mrs Ples (Australopithecus africanus, 2.5 million years old) — adds the deepest possible historical context to the most recent century's history.

Tunis is the most civilised city on the North African coast and the least known to British visitors, who overwhelmingly visit Morocco instead. This is a navigational error of some significance: Tunisia is smaller, safer and more immediately legible than Morocco, its cooking is the finest in North Africa, its historical sites include the ruins of Carthage (the most powerful city in the western Mediterranean before Rome destroyed it in 146 BCE), and its capital has an extraordinarily beautiful UNESCO-listed medina that is simultaneously the most intact medieval Islamic city in North Africa and the most genuinely inhabited — Tunisian families still living in the same houses their ancestors built in the 8th century, their courtyards fragrant with jasmine and orange blossom, their streets shaded by carved wooden lattice screens designed to allow women to observe the street without being observed.

The Medina of Tunis — inscribed on the UNESCO World Heritage List in 1979, covering 270 hectares within the old city walls — is structured around the Zitouna Mosque (the Great Mosque of the Olive Tree, founded in 698 CE, the oldest institution of Islamic learning in the western Islamic world, its columns recycled from Carthage and its surrounding streets still lined with the theological schools and scholars' workshops of a medieval university city). The specific flavour of the medina — its relative quietness compared with the Marrakesh or Fez souks, its alternation of grand dar mansions and small artisan workshops, its extraordinary concentration of decorative tiles in the Tunisian zellige tradition — makes it the most contemplative of North Africa's historic cities.

The ruins of Carthage, on a promontory 15km northeast of Tunis (accessible by TGM light rail, 40 minutes from the city centre), are among the most romantically situated archaeological sites in the Mediterranean: Phoenician harbour installations, Roman baths, a Tophet (child sacrifice precinct, the most controversial site in the ancient world), Byzantine churches and an amphitheatre, all in a residential suburb where villas overlook the Bay of Tunis and the ruins are distributed among gardens and viewpoints. The Bardo National Museum — the finest collection of Roman mosaics in the world, housed in a former Husainid palace — is the single most important reason to visit Tunis, and it remains, inexplicably, one of the least visited world-class museums in the Mediterranean.