This paper presents a complete realization theory for a class of discrete-time, bilinear systems with observed stochastic inputs, which we call generalized bilinear systems. This class of systems includes subclasses of bilinear systems, linear parameter varying (LPV) systems, and jump-Markov linear systems. We present necessary and sufficient conditions for the existence of a realization of generalized bilinear systems, along with a characterization of minimality in terms of rank conditions. We also formulate a realization algorithm and a minimization algorithm and we show that minimality can be checked algorithmically.