  We present a rather general method for proving local limit theorems, with a
good rate of convergence, for sums of dependent random variables. The method is
applicable when a Stein coupling can be exhibited. Our approach involves both
Stein's method for distributional approximation and Stein's method for
concentration. As applications, we prove local central limit theorems with rate
of convergence for the number of germs with $d$ neighbours in a germ--grain
model, and the number of degree-$d$ vertices in an Erd\H{o}s--R\'enyi random
graph. In both cases, the error rate is optimal, up to logarithmic factors.
