In this paper, we obtain necessary and sufficient conditions for the existence of diffeomorphisms that transform stochastic nonlinear systems to various canonical forms. The main tool in our analysis is the so called invariance under transformation rule that directly relates the coordinate transformation for stochastic nonlinear systems to that for deterministic uncertain nonlinear systems. This invariance rule allows the utilization of the existing necessary and sufficient conditions for deterministic nonlinear systems in associated stochastic nonlinear systems.