  We study the connection between amenability, F{\o}lner conditions and the
geometry of finitely generated semigroups. Using results of Klawe, we show that
within an extremely broad class of semigroups (encompassing all groups, left
cancellative semigroups, finite semigroups, compact topological semigroups,
inverse semigroups, regular semigroups, commutative semigroups and semigroups
with a left, right or two-sided zero element), left amenability coincides with
the strong F{\o}lner condition. Within the same class, we show that a finitely
generated semigroup of subexponential growth is left amenable if and only if it
is left reversible. We show that the (weak) F{\o}lner condition is a left
quasi-isometry invariant of finitely generated semigroups, and hence that left
amenability is a left quasi-isometry invariant of left cancellative semigroups.
We also give a new characterisation of the strong F{\o}lner condition, in terms
of the existence of weak F{\o}lner sets satisfying a local injectivity
condition on the relevant translation action of the semigroup.
