In this chapter we consider Markov Decision Processes with an infinite time horizon. There are situations where problems with infinite time horizon arise in a natural way, e.g. when the random lifetime of the investor is considered. However more important is the fact that Markov Decision Models with finite but large horizon can be approximated by models with infinite time horizon. The latter one is often simpler to solve and admits mostly a (time) stationary optimal policy. On the other hand, the infinite time horizon makes it necessary to invoke some convergence assumptions. Moreover, for the theory it is necessary that properties of the finite horizon value functions carry over to the limit function.