We study the cohomology of differential complexes, which we shall call Dolbeault-Kostant complexes, defined by certain integrable sub-bundles F of the complex tangent bundle of a manifold M . When M has a complex or symplectic structure and F is chosen to be the bundle of anti-holomorphic tangent vectors or, respectively, a “polarization” then the corresponding complexes are, respectively, the Dolbeault complex and (under further conditions) a complex introduced by Kostant in the context of geometric quantization. A simple condition on F insures that our complexes are elliptic. Assuming ellipticity and compactness of M , for example, one of our key results is a Hirzebruch-Riemann-Roch Theorem.