Concentration inequalities are very often a crucial step in deriving many results in statistical learning. The purpose of this talk is to present Bernstein and Hoeffding type maximal inequalities for regenerative Markov chains. Furthermore, we generalize these results and show exponential bounds for suprema of empirical processes over a class of functions F which size is controlled by its uniform entropy number. We show also that concentration inequalities are possible to obtain when the chain is sub-geometric. All constants involved in the bounds of the considered inequalities are given in an explicit form which can be advantageous in practical considerations. We show that the inequalities obtained for regenerative Markov chains can be easily generalized to a Harris recurrent case. Finally we provide one example of application of presented inequalities in statistical learning theory and obtain generalization bounds for mimimum volume set estimation problem when the data are Markovian.