*also affiliated with Jilin University The talk concerns stationary measures of diffusion processes generated from white-noise perturbed systems of ordinary differential equations. By relaxing the notion of Lyapunov functions associated with the stationary Fokker-Planck equations, new existence and non-existence results of stationary measures will be presented. As noises vanish, concentration and limit behaviors of stationary measures will be described with particular attentions paying to the special role played by multiplicative noises. Connections to problems such as stochastic stability and stochastic bifurcations will also be discussed.