Assume thatϕ(R, m, k) → (S, n, l) is a local homomorphism between commutative noetherian local ringsRandS. We say that anS-moduleMis almost finite overRif it is finitely generated overS(theR-structure onMis induced by ϕ). We investigate the homological behaviour of such modules, as well as various properties of the ringsRandSin the presence of an almost finite module of finite flat dimension overR.