S-operators were originally introduced by V. Lafforgue as certain operators acting on the cohomology of moduli of Shtukas. I will discuss their analogues and generalizations in the Shimura variety setting. They induce Hecke equivariant maps between cohomology of (different) Shimura varieties, and realize the generic Tate classes on the mod p fibers of Shimura varieties. Time permitting, I will move to the second part of the title and explain how they arise as the categorical trace of (a motivic version of) the geometric Satake. This is based on a joint work with Liang Xiao.