The Doge appoints a Council of Ten who exercise the powers of a supreme judicial court and executive advisory cabinet. Most of the time, the Council of Ten is the one that actually runs things, while the Doge smiles for the cameras.
The Signory is composed of most of the old mercantile and banking elites, as well as a few favored relatives of the sitting Doge, who has sole authority to appoint members. Tradition is that scions of the elite have thei names written in the Golden Book--the signory membeship roster-on the day of their birth.
painting at at the Grand Council Chamber