The intrinsic volumes of a convex body are fundamental invariants that capture information about the average volume of the projection of the convex body onto a random subspace of fixed dimension. The intrinsic volumes also play a central role in integral geometry formulas that describe how moving convex bodies interact. The purpose of this article is to derive concentration inequalities for the intrinsic volumes and related sequences using geometric methods. These concentration results complement recent concentration results for ultra log-concave sequences and have striking implications for high-dimensional integral geometry. In particular, they uncover new phase transitions in formulas for random projections, rotation means, random slicing, and the kinematic formula. In each case, the location of the phase transition is determined by reducing each convex body to a single summary parameter.