The processes with regime-switching can be used to characterize a stochastic dynamical system in a random environment. We investigate the long time behavior of such kind of processes including the recurrence and transience property, heavy-tail and light-tail property. We focus on two typical examples: Cox-Ingersoll-Ross process and Ornstein-Uhlenbeck process driven by Levy noise with regime-switching. We provide explicit criteria on the transience and recurrence of such processes. And when the stationary distribution exists, we provide sufficient conditions to justify whether this law is heavy-tailed or light-tailed.  We shall show that both the jumps caused by the Levy noise and regime-switching can derive the heavy-tailed property of the stationary distribution. Also, the different role played by Levy measure and regime-switching process is clearly characterized.

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