  This paper provides a short introduction to the notion of regular category
and its use in categorical algebra. We first prove some of its basic
properties, and consider some fundamental algebraic examples. We then analyse
the algebraic properties of the categories satisfying the additional Mal'tsev
axiom, and then the weaker Goursat axiom. These latter contexts can be seen as
the categorical counterparts of the properties of $2$-permutability and of
$3$-permutability of congruences in universal algebra. Mal'tsev and Goursat
categories have been intensively studied in the last years: we present here
some of their basic properties, which are useful to read more advanced texts in
categorical algebra.
