The gas-liquid membrane contactor (GLMC) is a promising technology for gas absorption and stripping with many competitive advantages over conventional contacting processes. At the centre of this technology, membranes are instrumental in offering a high contact area at the gas-liquid interface. Today, polymer-based membranes are the mainstream for GLMC applications owing to their well-established fabrication techniques and economic advantages. Herein, we aim to review the current progress in polymer-based membranes for GLMC processes. Specifically, we first provide a basic overview of the principles and background of GLMC. Next, we identify three of the biggest Achilles heels of polymer-based membranes for GLMC processes, namely, (1) membrane performances, (2) pore wetting, and (3) membrane fouling. To a certain extent, we contend that a large part of these challenges can be resolved by optimizing the membrane designs. As such, in the next section, we focus our attention on high-performance, antiwetting and antifouling membranes as potential solutions to address these challenges. From a membrane-centred perspective, the scope of our discussion covers extensively from membrane materials, morphologies, structures and configurations, surface modifications to novel membrane designs including composite and mixed-matrix membranes. Notably, as membrane fouling studies are still in the preliminary stage, attractive membrane designs such as Janus and omniphobic membranes are borrowed from membrane distillation process to gain greater insights into the potential solutions that can be adapted for GLMC processes.