Let S be a semigroup, and let l^1(S) be the Banach algebra which is the semigroup algebra on S. We shall determine when l^1(S) is amenable; the case where S is a group is the famous theorem of Barry Johnson. We shall also discuss constants of amenability. We shall move on to discuss the amenability and weak amenability of the second dual algebra l^1(S)'' - this is an algebra of measures on beta S.

[This is work with A. To-Ming Lau and D. Strauss.]