The well-known topological Riemann-Hilbert correspondence states that   there is an equivalence of categories between flat vector bundles on a smooth   manifold M and representations of the fundamental group of M. On the other   hand, the data of a flat connection on a bundle E is encoded in a twisted de   Rham complex \Omega(M, E). The latter is a differential graded complex over   the de Rham algebra \Omega(M) of a special kind. In this talk I will explain   what happens if we consider arbitrary \Omega(M)-modules and the corresponding   coderived category. This is joint work with J. Chuang and J. Holstein.