This paper is concerned with the following problem: Given a stochastic nonlinear system controlled over a noisy channel, what is the largest class of channels for which there exist coding and control policies so that the closed-loop system is stochastically stable? Stochastic stability notions considered are stationarity, ergodicity or asymptotic mean stationarity. We do not restrict the state space to be compact, for example, systems considered can be driven by unbounded noise. Necessary and sufficient conditions are obtained for a large class of systems and channels. A generalization of Bode's integral formula for a large class of nonlinear systems and information channels is obtained. The findings generalize existing results for linear systems.