Membranes for use in methods and apparatuses for hemodialysis and hemofiltration are composed of at least one polyarylethernitrile block copolymer having structural units of formula I ##STR00001## wherein Z is a direct bond, O, S, CH.sub.2, SO, SO.sub.2, CO, RPO, CH.sub.2, alkenyl, alkynyl, a C.sub.1-C.sub.12 aliphatic radical, a C.sub.3-C.sub.12 cycloaliphatic radical, a C.sub.3-C.sub.12 aromatic radical or a combination thereof; R is a C.sub.6-12 aromatic radical or a C.sub.1-12 aliphatic radical; R.sup.1 and R.sup.2 are independently H, halo, nitro, a C.sub.1-C.sub.12 aliphatic radical, a C.sub.3-C.sub.12 cycloaliphatic radical, a C.sub.3-C.sub.12 aromatic radical, or a combination thereof; a is 0, 1, 2 or 3; b is 0, 1, 2, 3 or 4; and m and n are independently 0 or 1.