During the last century, several function theories have been introduced over quaternions and over other alternative algebras. The idea behind these constructions is recovering in higher dimensions the refined tools available in the theory of holomorphic functions of one complex variable. The resulting theories, nonetheless, reflect the peculiar properties of the higher-dimensional algebras considered.

A relatively recent breakthrough was the definition, in 2006, of the notion of slice regular function of one quaternionic variable by Gentili and Struppa. This notion, generalized to alternative *-algebras by Ghiloni and Perotti in 2011, has rapidly grown into a full-fledged theory.

The talk will overview the general problem of function theory in one hypercomplex variable and the main features of the theory of slice regular functions, mentioning applications to open problems from other areas of mathematics.