A deployable membrane structure for an antenna comprises a membrane comprising a plurality of first regions of higher-stiffness material integrally connected via one or more second regions of lower-stiffness material, wherein the one or more second regions are formed from compliant material configured to permit the membrane to be folded into a collapsed configuration and subsequently unfolded into a deployed configuration, and are arranged so as to allow adjacent ones of the plurality of first regions to be folded so as to lie against one another. In some embodiments the membrane is formed of a composite material comprising a plurality of fibres in a compliant matrix, and the plurality of first regions comprise material with a higher fibre density than the one or more second regions. A deployable antenna comprising the deployable membrane structure is also disclosed.