  The goal of the memoir is to develop a new cohomology theory which
encompasses De Rham and Dolbeault cohomology as well as Deligne Beilinson
cohomology, in the context of general complex analytic manifolds. The special
case of the Iwasawa manifold is investigated as a typical example of what
occurs in the non K\"ahler case. Elementary applications to the Kodaira-Spencer
deformation theory and to the calculation of Chern classes are given.
