Graphene materials are increasingly being used in all types of membrane-based technologies ranging from water purification to energy conversion systems such as fuel cell and solar fuel technologies. Graphene's physical and chemical richness enables a variety of mass transport mechanisms that are impossible in traditional membranes. These mechanisms are dictated by the graphene microstructure and its versatility which leads transport to occur via pores, sub-nanometer pores, or via nanochannels. Graphene membranes can be designed to create nanochannels that allow highly selective ion or gas transport by biomimicking naturally occurring biological systems. However, these potentials cannot be fully explored without understanding the interplay between graphene microstructures and the transport mechanism. Toward taking advantage of graphene membrane technology, a fundamental understanding of the mass transport mechanisms is necessary. In this Review, we provide an in-depth discussion about three different types of mass transport through graphene nanopores and graphene nanochannels. By focusing on the relation between nanopores/channels chemistry e.g. effect of functional groups and pore polarity and transport mechanism, we identify key lessons, challenges, and potential solutions to empowering membrane-based energy technologies.