The National Museum is split across two adjacent 19th-century merchant houses on the slope below Saint Sophia. Together they hold the city's collection of medieval icons — one of the most important in the world outside Istanbul, with works from the 11th to the 14th century. The icons of the Annunciation and Christ the Saviour are the centrepieces.
The same site contains an archaeology section with Roman material from the surrounding lakeshore villas, and an ethnography section in the upper rooms. The icon collection alone justifies the visit. Allow at least an hour; longer if you read all the panels.