Stochastic differential equations (SDEs) are widely adopted to describe systems with stochastic disturbances, while they are not necessarily the best models in some specific situations. This paper considers the nonlinear systems described by random differential equations (RDEs). The notions and the corresponding criteria of noise-to-state stability, asymptotic gain and asymptotic stability are proposed, in the m-th moment or in probability. Several estimation methods of stochastic processes are presented to explain the reasonability of the assumptions used in theorems. As applications of stability criteria, some examples about stabilization, regulation and tracking are considered, respectively. A theoretical framework on stability of RDEs is finally constructed, which is distinguished from the existing framework of SDEs.