We deal with two themes that are illustrative of the rigidity and regularity of holomorphic mappings. The first one concerns the regularity of continuous CR mappings between smooth pseudo convex, finite type hypersurfaces which is a well studied subject for it is linked with the problem of studying the boundary behaviour of proper holomorphic mappings between domains bounded by such hypersurfaces. More specifically, we study the regularity of Lipschitz CR mappings from an h-extendible(or semi-regular) hypersurface in Cn .Under various assumptions on the target hypersurface, it is shown that such mappings must be smooth. A rigidity result for proper holomorphic mappings from strongly pseudo convex domains is also proved. The second theme dealt with, is the classification upto biholomorphic equivalence of model domains with abelian automorphism group in C3 .It is shown that every model domain i.e.,a hyperbolic rigid polynomial domainin C3 of finite type, with abelian automorphism group is equivalent to a domain that is balanced with respect to some weight.