The demand for large antennas in future space missions has increasingly stimulated the development of deployable membrane antenna structures owing to their light weight and small stowage volume. However, there is little literature providing a comprehensive review and comparison of different membrane antenna structures. Space-borne membrane antenna structures are mainly classified as either parabolic or planar membrane antenna structures. For parabolic membrane antenna structures, there are five deploying and forming methods, including inflation, inflation-rigidization, elastic ribs driven, Shape Memory Polymer (SMP)-inflation, and electrostatic forming. The development and detailed comparison of these five methods are presented. Then, properties of membrane materials (including polyester film and polyimide film) for parabolic membrane antennas are compared. Additionally, for planar membrane antenna structures, frame shapes have changed from circular to rectangular, and different tensioning systems have emerged successively, including single Miura-Natori, double, and multi-layer tensioning systems. Recent advances in structural configurations, tensioning system design, and dynamic analysis for planar membrane antenna structures are investigated. Finally, future trends for large space membrane antenna structures are pointed out and technical problems are proposed, including design and analysis of membrane structures, materials and processes, membrane packing, surface accuracy stability, and test and verification technology. Through a review of large deployable membrane antenna structures, guidance for space membrane-antenna research and applications is provided.