Cohomology theory for Banach algebras has enjoyed notable successes,
but -- outside the class of amenable algebras -- is somewhat short on
examples where one can compute the cohomology in degrees 2 and above.
We shall discuss some results for the class of l^1-semigroup algebras,
with particular focus on the commutative case where extra algebraic
machinery is available. The aim is to give an overview of both old and
recent work, using Clifford semigroups and free abelian semigroups as
the motivating examples. We present some recent results of the speaker
on both cases, and make some suggestions for future research.