A classical result due to Dobrushin (1956) yields the central limit theorem for partial sums of functionals of inhomogeneous (“sufficiently contracting”) Markov chains. In the talk we will restrict to bounded functionals of uniformly elliptic inhomogeneous Markov chains, for which we can obtain: A Berry-Esseen theorem (optimal rates in the Central limit theorem); Correction terms of order 1 in the CLT; Higher order correction terms in the CLT, commonly known as Edgeworth expansions. For Markov chains on compact Riemannian manifolds or for functional with decaying supremum norms we obtain optimal conditions for the Edgeworth expansions of an arbitrary order. We will also discuss the “universality of Edgeworth polynomials”.