Vector valued holomorphic functions

We survey results on holomorphic functions (of one complex variable) with values in a complex topological vector space hinting to their extension to the case of several complex variables. We give a version of the Hartogs theorem on separate analyticity for weakly holomorphic functions with values in a complex Fréchet space. The theory of $\alpha$-differentiable functions (due to N. Teodorescu, and extended by F-H. Vasilescu, to functions with values in a Fréchet space) is briefly reviewed as related to areolar derivatives. We present a selection of results on holomorphic functions with values in a complex Banach space with an emphasis on the boundary behavior of vector-valued holomorphic functions. We announce an extension of work by M.S. Baouendi & F. Treves, (on the approximation of CR functions by holomorphic functions) to the case of CR functions with values in a complex Fréchet space.