Abstract: A continuous time Markov chain X on non-negative integers is called discrete self-similar if for any p in the interval [0,1], it satisfies Binomial(X(t,n),p)=X(t,Binomial(n,p)) in distribution. In this work, we identify a large class of discrete self-similar Markov chains, which also includes the classical birth-death chains. We show that the semigroups associated to these Markov chains satisfy an intertwining relation with the semigroups of self-similar Markov processes on the positive real line. Because of this fact, these Markov chains converge to the self-similar Markov processes on the positive real line after scaling appropriately. By a linear perturbation of the generator of the discrete self-similar Markov chains, we obtain a class of ergodic Markov chains that are non-reversible. In this talk, I will explain how we obtain the spectral properties, entropy decay and hypercontractivity phenomenon of these Markov chains using the concept of intertwining and interweaving relationship. This is a joint work with Laurent Miclo (CNRS and University of Toulouse) and Pierre Patie (Cornell University).