  We derive explicit central moment inequalities for random variables that
admit a Stein coupling, such as exchangeable pairs, size--bias couplings or
local dependence, among others. The bounds are in terms of moments (not
necessarily central) of variables in the Stein coupling, which are typically
local in some sense, and therefore easier to bound. In cases where the Stein
couplings have the kind of behaviour leading to good normal approximation, the
central moments are closely bounded by those of a normal. We show how the
bounds can be used to produce concentration inequalities, and compare them to
those existing in related settings. Finally, we illustrate the power of the
theory by bounding the central moments of sums of neighbourhood statistics in
sparse Erd\H{o}s--R\'enyi random graphs.
