  In this paper we develop a theory of slice regular functions on a real
alternative algebra $A$. Our approach is based on a well--known Fueter's
construction. Two recent function theories can be included in our general
theory: the one of slice regular functions of a quaternionic or octonionic
variable and the theory of slice monogenic functions of a Clifford variable.
Our approach permits to extend the range of these function theories and to
obtain new results. In particular, we get a strong form of the fundamental
theorem of algebra for an ample class of polynomials with coefficients in $A$
and we prove a Cauchy integral formula for slice functions of class $C^1$.
