A new theory of generalized functions defined on open subsets of \({\mathbb{R}}^ n\) has been introduced by Colombeau; see his books ''New generalized functions and multiplication of distributions'' (1984; Zbl 0532.46019) and ''Elementary introduction to new generalized functions'' (1985). In this paper, the authors define the holomorphic generalized functions on an open subset \(\Omega\) of \({\mathbb{C}}^ n\) as those generalized functions G on \(\Omega\) that satisfy \({\bar \partial}G=0\). They start the study of these holomorphic generalized functions by proving that they have several properties of the usual holomorphic functions. Nevertheless they find also serious differences between these new ''functions'' and the classical ones, with respect to analytic continuation. The holomorphic generalized functions cannot be distributions except if they are already usual holomorphic functions. The paper is devoted mostly to one complex variable; holomorphic generalized functions of several complex variables are mentioned briefly at the end. In notes added in proof, it is stated that: (1) Holomorphic generalized functions are also considered in Chapter 8 of the first book mentioned precedingly, in a setting somewhat different from the one of this article. (2) The problem of the uniqueness of analytic continuation for a holomorphic generalized function null on an open nonvoid subset of a connected open set has been afirmatively solved by the two authors very recently.