The problem of model reduction for stochastic nonlinear systems is addressed with the moment matching method. We characterize the steady-state of nonlinear stochastic systems exploiting a stochastic version of the center manifold theorem. Exploiting the steady-state response we formulate the notion of moment for stochastic nonlinear systems and we solve the problem of model reduction proposing a family of nonlinear stochastic reduced order models. Moreover, we formulate also the notion of nonlinear stochastic reduced order model in the mean. The advantage of this last family of stochastic models is that the complexity of their determination is equal to the complexity of the determination of deterministic reduced order models. The relation between the two families of reduced order models is illustrated by means of a simple example based on an electrical circuit.