Icherisheher, the Old City, is a small walled quarter at the heart of Baku and the only one that gives any real sense of the medieval port that once stood on this site. It became a UNESCO World Heritage Site in 2000. Inside the walls you'll find the Maiden's Tower, the Palace of the Shirvanshahs, dozens of caravanserais turned into cafes and carpet shops, mosques both grand and modest, and the slightly disorienting experience of wandering between buildings that span eight centuries.
It's also where most visitors stay or eat at least once. The streets are stone, the cars are kept out of most of them, and the whole quarter empties out at night except for a handful of restaurants. Locals call it simply Içəri Şəhər.