This article deals with thickness in topological transformation semigroups
(τ-semigroups). Thickness is used to establish conditions guaranteeing an invariant mean on a
function space defined on a τ-semigroup if there exists an invariant mean on its functions
restricted to a sub-τ-semigroup of the original τ-semigroup. We sketch earlier results, then give
many equivalent conditions for thickness on τ-semigroups, and finally present theorems giving
conditions for an invariant mean to exist on a function space.