The richness of the theory of holomorphic functions of one complex variable arouse interest in "resembling" theories for quaternionic functions of a quaternionic variable. The first aim of the talk is to present the definition of (Cullen or) slice regular function over quaternions, accompanied by the main reasons that motivated it. The basic results of this recent theory will then be illustrated, paying particular attention to those topics in which the results are diverse from those which hold true for the theory of complex holomorphic functions.