In the last decade, there has been a growing trend towards flexible body wave energy converters (WECs) enabled by rubber-like elastomeric composite membrane structures that can simplify all aspects of WEC design. Currently, there are few literature studies detailing the implementations of membranes into WEC design. This paper aims to overcome this by reviewing the developments, material selection and modelling procedures for novel membrane based wave energy converters (mWECs), providing the reader with a comprehensive overview of the current state of the technology. In the first half of this paper, all of the possible WEC implementation areas are reviewed which include the primary mover, power take-off (PTO) and other sub-assembly systems. For the primary mover, the review has identified three main working surface approaches using membranes, these are: air-filled cells, water filled tubes and tethered carpets; which aim to reduce peak loads for enhanced reliability and survivability. In other areas, the PTO of WECs can benefit from using soft dielectric elastomer generators (DEGs) which offer a simpler designs compared with conventional mechanical turbomachinery. These have been implemented into the membrane working surface as well as replacing the PTO in existing WEC architectures. In the second half of the paper, a discussion is made on the material selection requirements with a few possible compositions presented. Following this, the potential modelling procedures for these devices is detailed. The device numerical models have altered existing procedures to take into account the non-linearities caused by the membrane interface and membrane PTO damping.