Let R be a commutative ring and E an R-module. Let \(f=(I_ n)\) be a filtration on R and \(e=(E_ n)\) a filtration on E. In the case where \(I_ n=I^ n\), I an ideal, e is said to be I-good if \(IE_ n\subseteq E_{n+1}\) for all \(n\geq 1\) with equality for large n. The purpose of this paper is to extend results concerning I-good filtrations to f-good filtrations where e is defined to be f-good if \(I_ mE_ n\subseteq E_{m+n}\) for all m,n\(\geq 1\) (i.e., e is f-compatible) and there exists an m such that \(E_ n=\sum^{m}_{i=1}I_{n-i}E_ i\quad for\) all \(n>m\). For example, it is shown that if each \(E_ n\) is finitely generated and e is f-compatible, then e is f-good if and only if E is a finitely generated \(S(R,f)=R[tI_ 1,t^ 2I_ 2,...]\)-module. This is used to obtain various analogs of the Artin-Rees lemma. Necessary and sufficient conditions are given for the Rees ring of f, \(R(R,f)=S[R,f][t^{-1}]\) to be Noetherian. Special attention is given to essentially power filtrations (e.p.f): f is an e.p.f. if there exists a positive integer m such that \(I_ n=\sum^{m}_{i=1}I_{n-i}I_ i\quad for\) all \(n\geq 1\), where \(I_ j=R\) for \(j\geq 0\).