  We systematically develop a theory of graded semigroups, that is semigroups S
partitioned by groups G, in a manner compatible with the multiplication on S.
We define a smash product S#G, and show that when S has local units, the
category S#G-Mod of sets admitting an S#G-action is isomorphic to the category
S-Gr of graded sets admitting an appropriate S-action. We also show that when S
is an inverse semigroup, it is strongly graded if and only if S-Gr is naturally
equivalent to S_1-Mod, where S_1 is the partition of S corresponding to the
identity element 1 of G. These results are analogous to well-known theorems of
Cohen/Montgomery and Dade for graded rings. Moreover, we show that graded
Morita equivalence implies Morita equivalence for semigroups with local units,
evincing the wealth of information encoded by the grading of a semigroup. We
also give a graded Vagner-Preston theorem, provide numerous examples of
naturally-occurring graded semigroups, and explore connections between graded
semigroups, graded rings, and graded groupoids. In particular, we introduce
graded Rees matrix semigroups, and relate them to smash product semigroups. We
pay special attention to graded graph inverse semigroups, and characterise
those that produce strongly graded Leavitt path algebras.
