Innsbruck, Austria

Innsbruck is the capital of Tyrol and the most fully alpine of Austria's cities — set on the Inn river floor with the limestone Karwendel range rising vertically above the northern bank and the Patscherkofel facing it across the valley. The city's defining trick is the speed at which it changes scales: a ten-minute walk and a single cable-car ride take you from the Goldenes Dachl in the Habsburg old town to the Hafelekar summit at 2,256 metres, with the Nordkette Zaha Hadid stations along the route. Innsbruck twice hosted the Winter Olympics (1964 and 1976) and is one of very few cities where serious skiing, ski touring and alpine climbing begin inside the urban transit network. Beyond the alpine identity, the Habsburg heritage runs deep: the Hofkirche holds Maximilian I's monumental cenotaph surrounded by 28 over-life-size bronze figures, the Hofburg state apartments rival their Vienna counterparts in scale, and the Schloss Ambras Renaissance collection is one of the world's most important.