Partial actions of groups and semigroups

We define the notion of a partial action of a group G on a set X. Partial actions of groups were formally introduced by Exel in the 90's in the context of partial actions of groups on associative algebras. Exel, Kellendonk and Lawson later showed that every partial action of a group G on a set X can be replaced by an action of an inverse semigroup Sz(G) on X. Here Sz(G) is an expansion of G. On the other hand, Kellendonk and Lawson have shown that a group G acts partially on a set X if and only if a globalisation exists, that is, there is a set Y containing X such that the partial action of G on X is the restriction to X of an action of G on Y ^Ö thus confirming the naturalness of the notion of partial action. A group is certainly a special kind of inverse semigroup. What then of partial actions of inverse semigroups, or indeed of semigroups from wider classes? We discuss how such partial actions are defined, and the relevant expansion and globalisation results. The expansion results are intimately connected with the notion of F-morphism and the results of Lawson, Margolis and Steinberg, and those of Gomes.