  For any reductive group G over a global function field, we use the cohomology
of G-shtukas with multiple modifications and the geometric Satake equivalence
to prove the global Langlands correspondence for G in the "automorphic to
Galois" direction. Moreover we obtain a canonical decomposition of the spaces
of cuspidal automorphic forms indexed by global Langlands parameters. The proof
does not rely at all on the Arthur-Selberg trace formula.
