The main aim of this thesis is to generalise McAlister's the-ory of locally inverse regular semigroups to the class of semigroups with local units in which t he iocal submonoids have commuting idempotents. We prove that if such a semigroup has what we call a McAlister sandwich function then the semigroup can be covered by means of a Rees matrix semigroup over a semigroup with com-muting idempotents. Examples of such semigroups are easily con-structed. Indeed, if T is a semigroup with local units having an idempotent e such that T = TeT, and eTe has commuting idem-potents, then all t he local submonoids of T have commuting idem-potents and T is equipped with a McAlister sandwich function. We prove that the semigroups with local units having local submonoids with commuting idempotents S which can be embedded in such a semigroup T in such a way that S = ST S are precisely the ones having a McAlister sandwich function. Finally, in a different direction, we study variants of semigroups concentrating on the relationship between the local structure of a semigroup and the global structure of its variants.