Graphene is a well-known two-dimensional material that exhibits preeminent electrical, mechanical and thermal properties owing to its unique one-atom-thick structure. Graphene and its derivatives (e.g., graphene oxide) have become emerging nano-building blocks for separation membranes featuring distinct laminar structures and tunable physicochemical properties. Extraordinary molecular separation properties for purifying water and gases have been demonstrated by graphene-based membranes, which have attracted a huge surge of interest during the past few years. This tutorial review aims to present the latest groundbreaking advances in both the theoretical and experimental chemical science and engineering of graphene-based membranes, including their design, fabrication and application. Special attention will be given to the progresses in processing graphene and its derivatives into separation membranes with three distinct forms: a porous graphene layer, assembled graphene laminates and graphene-based composites. Moreover, critical views on separation mechanisms within graphene-based membranes will be provided based on discussing the effect of inter-layer nanochannels, defects/pores and functional groups on molecular transport. Furthermore, the separation performance of graphene-based membranes applied in pressure filtration, pervaporation and gas separation will be summarized. This article is expected to provide a compact source of relevant and timely information and will be of great interest to all chemists, physicists, materials scientists, engineers and students entering or already working in the field of graphene-based membranes and functional films.