Hecke operators are an indispensable tool to understand the structure of spaces of classical modular forms, and more generally of the cohomology of automorphic vector bundles on Shimura varieties. In order to obtain a good integral theory, the naive definition of Hecke operator needs to be normalized in the right way. This phenomenon is visible already for modular forms, where the normalization may appear a little artificial. The correct correction term can be found, also more generally, using the geometric construction of Hecke operators by correspondences, but involves the geometry of the special fiber in subtle and interesting ways. While the case of modular forms and modular curves is classical, integrality properties of certain Hecke operators were studied for other Shimura varieties by Fakhrudin and Pilloni. In the talk, I will explain the principle of the argument and how to apply it to a significantly larger class of Hecke operators. This is joint work with Xuhua He and Michael Rapoport.