This talk will be on two themes that illustrate the rigidity and regularity of holomorphic mappings. The first part will deal with 
 results concerning the smoothness of continuous CR (Cauchy -- Riemann) mappings; in particular, that of Lipschitz continuous 
 CR mappings from h-extendible/semi-regular hypersurfaces into certain Levi co-rank one hypersurfaces, in C^n. The second part 
 will deal with the classification of Kobayashi hyperbolic, finite type rigid polynomial domains in C^3.