Abstract: Let $p$ be a prime number. Let $(G, b, \mu)$ be a local Shimura datum and $\CG$ be a quasi-parahoric group scheme for $G$ (these terms will be explained in the talk). Scholze has defined a functor on the category of perfectoid spaces in characteristic $p$ and has conjectured that this functor is representable by a formal scheme. I will explain the proof of this conjecture for classical groups. The conjecture also gives a group-theoretical interpretation of Rapoport-Zink spaces. This is joint work with G. Pappas.