We show that, providing a metric space $X$ has a boundary that is in somesense similar to the boundary of hyperbolic space, the iterates of acontraction $f:X\to X$ converge locally uniformly to a point in, or on theboundary of, $X$. This generalises the Denjoy–Wolff theorem for analyticself-maps of the unit disc in the complex plane, and also shows that if $D$is a bounded strictly convex subdomain of ${\Bbb R}^n$, then any contractionof $D$ with respect to the Hilbert metric of $D$ converges to a point in the closure of $D$.