The Dornier Museum sits out by the airport, a short bus ride from the centre, and is a serious aviation museum in its own right. Claude Dornier set up his aircraft manufacturing works here on the Bodensee after World War I, and the museum traces that history through original aircraft and detailed reconstructions. The collection includes the Do X flying boat — once the largest aircraft in the world — the Do 17 "flying pencil" bomber, and the Do 31 experimental VTOL jet transport. Outside on the apron you can board several aircraft. The building itself, a striking glass and steel structure, is worth looking at before you even go inside.
Good combination with the airport if you're arriving or departing by air; the museum is a five-minute walk from the terminal.