As we know, many important works in ergodic theory of stochastic dynamical systems have been obtained for invariant measures and aperiodic stationary processes. However, there exist non-stationary but ergodic processes. I will discuss the ergodic theory of three kind of non-stationary stochastic processes, which are random periodic processes, random quasi-periodic processes and stochastic processes under nonlinear expectation spaces. My talk will mainly concentrate on random periodic processes and periodic measures. This talk is based on some joint work with Y. J. Liu, Y. Liu, Y. J. Liu, Baoyou Qu, Huaizhong Zhao and Johnny Zhong.

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