The aim of this talk is to give a self-contained introduction to a new theory of regular functions over quaternions and, more in general, over a non commutative algebra. In particular, we’ll give a (historic) overview of different possible approaches to a definition of regularity for quaternionic functions and focus our  attention on the geometric/analytic properties related to the class of functions which are regular according to the recent definition given by Gentili & Struppa in 2007.

We’ll also present some applications and results which are in some cases a generalization of the similar results in Complex Analysis, in others are quite unexpected and promising for the study of new phenomena. Some related topics will be then considered in order to show what kind of  difficulties or open problems are still under investigation.