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Abstract. We define the notion of singular support of a coherent sheaf on a quasi-smooth derived (DG) scheme or Artin stack, where "quasi-smooth" means that it is a locally complete intersection in the derived sense. This develops the idea of "cohomological" support of coherent sheaves on a locally complete intersection scheme introduced by D. Benson, S. B. Iyengar, and H. Krause. We study the behaviour of singular support under the direct and inverse image functors for coherent sheaves.We use the theory of singular support of coherent sheaves to formulate the categorical Geometric Langlands Conjecture. We verify that it passes natural consistency tests: it is compatible with the Geometric Satake equivalence and with the Eisenstein series functors. The latter compatibility is particularly important, as it fails in the original "naive" form of the conjecture.

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