  We prove a Berry-Esseen theorem and Edgeworth expansions for partial sums of
the form $S_N=\sum_{n=1}^{N}f_n(X_n,X_{n+1})$, where $\{X_n\}$ is a uniformly
elliptic inhomogeneous Markov chain and $\{f_n\}$ is a sequence of uniformly
bounded functions. The Berry-Esseen theorem holds without additional
assumptions, while expansions of order $1$ hold when $\{f_n\}$ is irreducible,
which is an optimal condition.
  For higher order expansions, we then focus on two situations. The first is
when the essential supremum of $f_n$ is of order $O(n^{-\be})$ for some
$\be\in(0,1/2)$. In this case it turns out that expansions of any order
$r<\frac1{1-2\be}$ hold, and this condition is optimal. The second case is
uniformly elliptic chains on a compact Riemannian manifold. When $f_n$ are
uniformly Lipschitz continuous we show that $S_N$ admits expansions of all
orders. When $f_n$ are uniformly H\"older continuous with some exponent
$\al\in(0,1)$, we show that $S_N$ admits expansions of all orders
$r<\frac{1+\al}{1-\al}$. For H\"older continues functions with $\al<1$ our
results are new also for uniformly elliptic homogeneous Markov chains and a
single functional $f=f_n$. In fact, we show that the condition
$r<\frac{1+\al}{1-\al}$ is optimal
  even in the homogeneous case.
