This paper describes algorithms for computing the structure of finite transformation semigroups. The algorithms depend crucially on a now data structure for an R-class in terms of a group and an action. They provide for local computations, concerning a single R-class, without computing the whole semigroup, as well as for computing the global structure of the semigroup. The algorithms have been implemented in the share package MONOID within the GAP system for computational algebra. (C) 2002 Academic Press.