Recurrent motions play an important role in the study of differential equations and dynamical systems.Almost periodic (AP) and almost automorphic (AA) motions are recurrent motions that are extensively studied.In the theory of stochastic processes,the ``recurrence" has different but essentially similar meaning in some sense.In this talk,the AP processes will be recalled and AA processes will be introduced,both in sense of dynamical systems,to study stochastic differential equations.The concept of Poisson almost automorphy is introduced.The existence and uniqueness of AP or AA solutions to some linear and semilinear stochastic differential equations with infinite dimensional Levy noise are established provided the coefficients satisfy some suitable conditions.The global asymptotic stability of the unique AP or AA solution is discussed.