Membrane-based separation is a superior alternative to conventional processes in many separation problems of practical importance. For maximum effectiveness, both high selectivity and flux are desirable. This article summarizes the global efforts at designing new membrane materials, particularly carbon nanotube (CNT)-based membranes, to achieve the twin objectives mentioned above. Interest in CNT emanates from the excellent transport property of molecules through its frictionless smooth walls, with/without functionalization at the end tips. Permeation of water through graphene oxide (GO) channels, which are otherwise impermeable to solutes, has also generated considerable interest, and GO is being viewed as a promising material for separations. The performances of conducting polymers like polypyrrole, polyaniline, polythiophene-3, 4-ethylenedioxythiophene and their different composites have been studied as a function of changes in morphology and redox behaviour, and this aspect too is covered in the present article. Also, the dependence of the separation performances based on the size, charge and hydrophobic/hydrophilic properties has been discussed in detail. Advances in understanding will have an important bearing on future developments in separation science.