Many things happen at the Senate for while it is a place where civil matters are discussed and decided, it is also a place where business deals are made. Membership in the Senate proper is for a class of citizen know as "Senators" who represent the votes of Patricians and Equestrians at 50 and 10 to one respectively.
The Senate elected a leader called the, "First Among Equals," a position currently occupied by Vatha.
There is a path to citizenship but it involves gaining renown points, basically doing favors for the Senate or important senators.