Old problems in geometric probability have received renewed attention due to applications in optimization and statistics. Some of these applications are centred around the notion of intrinsic volumes, fundamental geometric invariants that include the Euler characteristic and the volume. Important results in integral geometry relate the intrinsic volumes of random projections, intersections, and sums of convex bodies to those of the individual volumes. We present a new interpretations of classic results, based on the observation that intrinsic volumes (both in spherical and Euclidean settings) concentrate around certain indices. I will give an overview of some old and new developments in this direction, and discuss further applications, for example to the convergence analysis of randomized algorithms. This is joint work with Joel Tropp.