Peer-to-peer, community or collective self-consumption, and transactive energy markets offer new models for trading energy locally. Over the past five years, there has been significant growth in the amount of academic literature examining how these local energy markets might function. This systematic literature review of 139 peer-reviewed journal articles examines the market designs used in these energy trading models. A modified version of the Business Ecosystem Architecture Modelling framework is used to extract market model information from the literature, and to identify differences and similarities between the models. This paper examines how peer-to-peer, community self-consumption and transactive energy markets are described in current literature. It explores the similarities and differences between these markets in terms of participation, governance structure, topology, and design. This paper systematises peer-to-peer, community self-consumption and transactive energy market designs, identifying six archetypes. Finally, it identifies five evidence gaps which require future research before these markets could be widely adopted. These evidence gaps are the lack of: consideration of physical constraints; a holistic approach to market design and operation; consideration about how these market designs will scale; consideration of information security; and, consideration of market participant privacy.