Turin

Begun as a synagogue, the spire now houses the National Cinema Museum. A glass lift shoots up through the centre of the exhibition hall to a terrace at 85m. The lift needs its own ticket, which sells out fast at weekends.

Opened in 1824, it is often ranked second only to Cairo for Egyptian holdings. Bread and linen from the tomb of the architect Kha and his wife survive intact. Tickets carry an entry slot, and booking one spares you the queue.

The Savoy palace, where one ticket carries on into the Royal Armoury and the Shroud Chapel. Guarini's chapel burned black in 1997 and did not reopen until 2018. The armoury is a long hall of armoured riders in a row.

The cathedral keeps the linen shroud said to have wrapped the body of Jesus. It is unrolled only for expositions held every few years; otherwise you see just its case in the left-hand chapel. Roman gate ruins run beside the church.

Two near-identical churches guard the south end, and old cafes fill the arcades down both sides. A bull is set in the pavement outside Caffe Torino; spinning on one heel over its private parts is supposed to bring good fortune, so that spot is worn hollow.

A Baroque basilica on the hilltop east of the city, with Savoy royal tombs below. On a sharp, cloudless day the Alps line up across the view from its forecourt. Behind the church is a memorial to the Torino football team whose plane hit this hill in 1949.

A long park beside the Po holds a castle and a medieval village. The village was built new for the 1884 exhibition, copying old Piedmont houses, so for all its age it is not truly medieval. Walking its lanes costs nothing.

The Savoy hunting palace, reached by bus out to the northern suburbs. Juvarra's Great Gallery is all white stucco, with light running its whole length. The gardens are wide enough that half a day goes quickly.

An open-air market covering all of Piazza della Repubblica, one of Europe's biggest. Vegetable stalls share it with indoor meat halls, and stalls are full only in the morning. On Saturday the Balon flea market spreads through the lanes to the north.

Fiat built cars in this old factory, and its oval test track still loops across the roof. The track is now a garden you can walk all the way round. The way up to the roof leads through the Pinacoteca Agnelli building.

The palace in Piazza Castello wears a baroque front Juvarra added from 1718, but round the back the towers of a medieval castle survive. Inside is the civic museum of ancient art, with the base of a Roman gate visible below the floor.

Guarini rippled the red-brick front of this palace, where the first Italian parliament convened in 1861. The museum inside tells the story of unification. The old ballroom used by the Sardinian kingdom's deputies from 1848 survives intact on the visitor route.

A 16th-century friary church on a hill across the Po, whose forecourt lines up river, rooftops and Alps in one view. Next door is the National Mountain Museum, founded by the Italian Alpine Club in 1874. People come up at dusk to see the Mole light up.

Built for King Victor Emmanuel I's return in 1814 after Napoleon's fall, under a dome copied from the Pantheon. Faith, a statue by the steps holding a chalice, is rumoured to gaze toward the hidden Holy Grail, and many visitors follow her line of sight.

Fiat's home town keeps its car museum on the Po, with nearly two hundred vehicles from late 19th-century steam carriages to racing cars. A 2011 redesign set them out like stage scenery. Lingotto lies a short stroll toward the river, an easy double bill.

The Savoy hunting lodge Juvarra began in 1729 on open land southwest of the city. A bronze stag on the central dome makes it easy to spot from afar. The straight tree-lined avenue running back toward town was the king's road to the hunt.

Abbey raised from the late 10th century on a crag where the Susa valley opens, the emblem of Piedmont. Umberto Eco reportedly pictured it for The Name of the Rose. The Staircase of the Dead leads inside; hiking up from the village takes ninety minutes or more.