I will talk about the random periodic solutions of random dynamical systems generated by stochastic differential equations and stochastic partial differential equations. I will start with definition and motivations of studying a random periodic solution, and discuss its connection with periodic measure. I will demonstrate that to find a random periodic solution for a hyperbolic system is equivalent to solve a solution of coupled infinite horizon stochastic integral equations. This works for non-dissipative stochastic systems. We then discuss mathematical tools, mainly Wiener-Sobolev compact embedding, to solve such coupled infinite horizon integral equations. This talk is based on a number of works, joint mainly with Z. Zheng and C. Feng respectively.