We give a geometric analog of the Weil representation of the metaplectic group, placing it in the framework of the geometric Langlands program. For a smooth projective curve $X$ we introduce an algebraic stack Bun$\backsim$G of metaplectic bundles on $X$. We give a tannakian description of the Langlands dual to the metaplectic group. Namely, we introduce a geometric version Sph of the (nonramified) Hecke algebra of the metaplectic group and describe it as a tensor category. The tensor category Sph acts on the derived category D(Bun$\backsim$G) by Hecke operators. Further, we construct a perverse sheaf on Bun$\backsim$G corresponding to the Weil representation and show that it is a Hecke eigensheaf with respect to Sph.