In our recent work, we showed that smooth CR-diffeomorphisms of real-analytic Levi-nonflat hypersurfaces in C^2 are not analytic in general. This result raised again the question on the nature of CR-maps of real-analytic hypersurfaces. In this paper, we give a complete picture of what CR-maps actually are. First, we discover an analytic continuation phenomenon for CR-dieomorphisms which we call the sectorial analyticity property. It appears to be the optimal regularity property for CR-dieomorphisms in general. We emphasize that such type of extension never appeared previously in the literature. Second, we introduce the class of Fuchsian type hypersurfaces and prove that (innitesimal generators of) CR-automorphisms of a Fuchsian type hypersurface are still analytic. In particular, this solves a problem formulated earlier by Shafikov and the first author. Finally, we prove a regularity result for formal CR-automorphisms of Fuchsian type hypersursufaces.