It is well-known that on a non-projective complex manifold, a
coherent sheaf may not have a resolution by a complex of holomorphic vector
bundles. Nevertheless, J. Block showed that such resolution always exists if
we allow anti-holomorphic flat superconnections which generalize complexes
of holomorphic vector bundles. Block's result makes it possible to study
coherent sheaves with differential geometric and analytic tools. For
example, in a joint work with J.M Bismut, S, Shen, and I, we give an
analytic proof of the Grothendieck-Riemann-Roch theorem for coherent sheaves
on complex manifolds. In this talk I will present the ideas and applications
of anti-holomorphic flat superconnections. I will also talk about analogous
constructions of superconnections in other areas of geometry.