In this paper we describe a novel approach to the study of U-statistics in the markovian setup, based on the (pseudo-) regenerative properties of Harris Markov chains. Exploiting the fact that any sample path X1, . . . , Xn of a general Harris chain X may be divided into asymptotically i.i.d. data blocks B1, . . . , BN of ran- dom length corresponding to successive (pseudo-) regeneration times, we introduce the notion of regenerative U-statistic ΩN =