This paper summarizes the recent progress on the limiting behavior of multiscale stochastic systems with irregular coefficients, focuses on presenting the averaging principle, the normal deviation and the diffusion approximation, in particular, the Smoluchowski-Kramers approximations for the classical stochastic Langevin equation driven by Brownian noise and the Langevin equation driven by L\'evy noise. Unlike the classical equation driven by Brownian noise, there is no noise induced drift in the limit equation of the Langevin euqation driven by L\'evy noise even if the friction is state dependent.

Diffusion approximation, Smoluchowski-Kramers approximation, Normal deviations, Multiscale system, Aeraging principle