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We propose a notion of algebra of {\it twisted} chiral differential operatorsover algebraic manifolds with vanishing 1st Pontrjagin class. We show that suchalgebras possess families of modules depending on infinitely many complexparameters, which we classify in terms of the corresponding algebra of twisteddifferential operators. If the underlying manifold is a flag manifold, ourconstruction recovers modules over an affine Lie algebra parameterized by opersover the Langlands dual Lie algebra. The spaces of global sections of"smallest" such modules are irreducible $\ghat$-modules and all irreducible$\frak{g}$-integrable $\ghat$-modules at the critical level arise in this way.

Algebras of twisted chiral differential operators and affine localization of $\frak{g}$-modules