Stuttgart

Ride up by lift to the highest floor first, then walk back down along two routes wound into a double helix. One route runs through the eras and the other through the vehicle types, and you can cross between them as you go. It opened in 2006.

It stands by the factory in Zuffenhausen to the north, the whole exhibition block floating on three pillars. Cars line ramps through a white interior, and a lower-floor window shows the workshop where old cars are restored. The train station is right outside.

A memorial column crowned by a goddess stands in the square fronting the New Palace. Sitting or lying on the broad lawns is allowed, and sunny days fill them with people out with lunch. When it rains, take the long Konigsbau colonnade on the north edge.

Built in 1956 and 217 metres tall, it became the model for TV towers in reinforced concrete. Because it rises from a wooded hill south of the centre, its deck surveys the entire basin, and in clear weather the ridge of the Swabian Jura too.

A royal garden laid out in Moorish style now serves as zoo and botanical garden. Old iron-and-glass hothouses and pillared walks survive among the animal enclosures. In April the central magnolia grove blooms all at once, and visitor numbers peak then.

A white cube by the Korean architect Eun Young Yi, opened in 2011. The top reading room is a white stepped space that narrows downward like a funnel, and the roof terrace above it is open to anyone who wants to look out over the city.

Housing built for a 1927 architecture show, led by Mies van der Rohe with each architect putting up a house. The two by Le Corbusier are World Heritage, and visitors may tour the semi-detached pair, now a museum. People still live in the rest.

King Wilhelm I built this domed chapel on the old castle hilltop for Queen Katharina, who died young. She was a Russian grand duchess, so it follows the Russian Orthodox style. Climbing from the village below through the vineyards is roughly a half-hour walk.

An Art Nouveau building from 1914, with fruit, spice and cheese stalls packed under a high glass roof. From the upper gallery you see the whole market floor below. It shuts on Sundays, so go on a Saturday morning.

More than 400 public stairways climb between the houses on the basin slopes, and locals call them Staffele. Pick one on the slope south of the centre and within minutes the city opens out over the rooftops. The climbs are long, so many plan the route downhill.

Beside the 19th-century gallery, James Stirling's new wing opened in 1984, pink and blue railings and green rubber floors set against stone. It is a landmark of postmodern architecture. A public path climbs through the round courtyard, no ticket needed.

About 20 minutes north by S-Bahn, one of Germany's largest baroque palaces, built over nearly 30 years in the early 18th century. Interiors are by guided tour only. Its Blooming Baroque gardens hold a fairy-tale garden and, every autumn, a pumpkin show.

A rococo hunting retreat Duke Carl Eugen completed on the western heights in 1769. A ruler-straight 13-kilometre avenue links it to Ludwigsburg palace and later served as the baseline for surveying Wurttemberg. The terrace looks far over the northern plain.

A 40-metre tower raised in 2001 in Killesberg park, a former quarry turned garden. Engineer Jorg Schlaich held it up with a net of cables, so you feel it sway slightly in the wind on the way up. The top shows the vineyard hills that ring the city.

A riverside old town 15 minutes away by train, its half-timbered houses largely spared by the war. An astronomical clock is set into the Old Town Hall front. In winter its medieval Christmas market, traders in period dress, is less crowded than Stuttgart's.

Stuttgart is said to hold Europe's largest mineral water reserves after Budapest, and this Bad Cannstatt pool fills straight from them. Outdoor basins stay open all winter, letting you sit in steaming water as snow falls. Iron makes the water slightly cloudy.

King Wilhelm I began it in 1818 as a farm fair after famine. It fills the Cannstatter Wasen in late September and early October; its emblem is a tall column of stacked fruit and vegetables. Beer tents fill on evenings and weekends, so go a weekday afternoon.