Abstract: In this talk we discuss the quantum Markov chains initiated by Accardi and Koroliuk. It was developed to extend the classical Markov chains to noncommutative spaces. Specifically we will consider the open quantum random walks in the view point of quantum Markov chains. Then we discuss the reducibility and irreducibility of open quantum random walks. Some examples will be provided. In particular, the classical Markov chains will be considered as open quantum random walks, and hence as quantum Markov chains. This talk is based on the joint work with Ameur Dhahri and Chul Ki Ko.