Let $S$ be a (discrete) semigroup, and let $\ell^{\,1}(S)$ be the Banach algebra which is the semigroup algebra of $S$. The authors study the structure of this Banach algebra and of its second dual. The authors determine exactly when $\ell^{\,1}(S)$ is amenable as a Banach algebra, and shall discuss its amenability constant, showing that there are 'forbidden values' for this constant. Table of Contents: Introduction; Banach algebras and their second duals; Semigroups; Semigroup algebras; Stone-?ech compactifications; The semigroup $(\beta S, \Box)$; Second duals of semigroup algebras; Related spaces and compactifications; Amenability for semigroups; Amenability of semigroup algebras; Amenability and weak amenability for certain Banach algebras; Topological centres; Open problems; Bibliography; Index of terms; Index of symbols. (MEMO/205/966)