In this talk, I will present our works giving an affirmative answer to Gromov's dihedral rigidity conjecture.  As a corollary, we obtain a positive answer to the Stoker problem, showing that the dihedral angles of a convex polyhedron determine its face angles. The key ingredient is the index theory of Dirac-type operators on polyhedrons, including essential self-adjointness and an index formula. This talk is based on joint works with Zhizhang Xie and Guoliang Yu.