DescriptionThe thesis consists of two parts. In the first part, we study a regularity problem for CR mappings between CR manifolds. More precisely, we establish various versions of the Schwarz reflection principle in several complex variables. In particular, as a consequence of the main results, we confirm a conjecture of X. Huang in [Hu2] and provide a solution to a question raised by Forstneric [Fr1] (See Corollaries 2.1.11 and 2.1.12). It is a joint work with Shiferaw Berhanu ([BX1], [BX2]). In the second part, we study the embeddability problem from compact real algebraic strongly pseudoconvex hypersurfaces into a sphere. In a joint work with Xiaojun Huang and Xiaoshan Li ([HLX]), we prove that for any integer $N,$ there is a family of compact real algebraic strongly pseudoconvex hypersurfaces in $mathbb{C}^2,$ none of which can be locally holomorphically embedded into the unit sphere in $mathbb{C}^N.$ This shows that the Whitney (or Remmert, respectively) type embedding theorem in differential topology (or in the Stein space theory, respectively) does not hold in the setting above