  We establish presumably optimal rates of normal convergence with respect to
the Kolmogorov distance for a large class of geometric functionals of marked
Poisson and binomial point processes on general metric spaces. The rates are
valid whenever the geometric functional is expressible as a sum of
exponentially stabilizing score functions satisfying a moment condition. By
incorporating stabilization methods into the Malliavin-Stein theory, we obtain
rates of normal approximation for sums of stabilizing score functions which
either improve upon existing rates or are the first of their kind. Our general
rates hold for functionals of marked input on spaces more general than
full-dimensional subsets of $\mathbb{R}^d$, including $m$-dimensional
Riemannian manifolds, $m\leq d$. We use the general results to deduce improved
and new rates of normal convergence for several functionals in stochastic
geometry, including those whose variances re-scale as the volume or the surface
area of an underlying set. In particular, we improve upon rates of normal
convergence for the $k$-face and $i$th intrinsic volume functionals of the
convex hull of Poisson and binomial random samples in a smooth convex body in
dimension $d\geq 2$. We also provide improved rates of normal convergence for
statistics of nearest neighbors graphs and high-dimensional data sets, the
number of maximal points in a random sample, estimators of surface area and
volume arising in set approximation via Voronoi tessellations, and clique
counts in generalized random geometric graphs.
