An irreducible Markov chain with a complete separable metric space of states is considered. In the class of all positive recurrent chains (in terms of Nummelin’s decomposition of an \(m\)-step transition function \((m \geq 1))\) the Nummelin’s method of regenerative approach is used for proving this chain to have a stationary version. The rate of convergence of one-dimensional distributions to a stationary one is estimated. In the class of all aperiodic chains, which uniformly quickly reach some set of typical states, continuity of finite-dimensional distributions with respect to weak convergence of one-step transition functions is proved. A special metric of weak convergence is used for the estimation of the modulus of continuity. Some conditions are proved to be sufficient for a sequence of Markov chains with finite state spaces to approximate the initial Markov chain. A distribution of a rare event is proved to be asymptotically exponential where the event corresponds to a sequence of nonempty sets with empty intersection.