It is expected that the global Langlands correspondence will be realized in the cohomology of Shimura varieties. This relies on the idea that these varieties are moduli for motives (and therefore in particular are sources for Galois representations), and in addition, they carry Hecke symmetries. A central point of Langlands' philosophy is that the zeta functions attached to Shimura varieties should be among the L-functions attached to automorphic representations of reductive groups. According to this, the conjecture of Langlands and Rapoport was born (~1987) to describe the structure of the points on the reduction of a Shimura variety. Recall that there is an analogues picture over function fields. Here, the moduli stacks for global G-shtukas replace Shimura varieties. In this talk, we first survey the panorama over function fields by translating the Deligne's conception of Shimura varieties (as a moduli for motives). Then we review the local theory of global G-shtukas, and continue towards the function fields analog of the conjecture of Langlands and Rapoport ...