This talk consists of two parts. In the first part, we exploit the connection between Markov chains and Stein’s method via the generator approach and express the solution of Stein’s equation in terms of expected hitting time. This yields new upper bounds of Stein’s factors in terms of the parameters of the Markov chains. In the second part, we study two types of Metropolis-Hastings (MH) reversiblizations for non-reversible Markov chains with transition kernel P. While the first type is the classical Metropolised version of P, we introduce a new self-adjoint kernel which captures the opposite transition effect of the first type, that we call the second MH kernel. This new approach has four attractive features which will be explained in details at the talk.