The production of cold antihydrogen atoms at CERN's Antiproton Decelerator (AD) has opened up the possibility to perform direct measurements of the Earth's gravitational acceleration on antimatter bodies. This is the main goal of the AEgIS, ALPHA-g and GBAR collaborations: to measure the value of g, using different approaches. These approaches, both continuous and pulsed, as well as a first result by the ALPHA-g collaboration, will be presented and discussed. Further physics directions also related to inertial sensing, in particular of the AEgIS collaboration relying on pulsed formation and manipulation processes, will also be touched upon, as they also open up unexplored venues in nuclear physics and in novel precision tests of fundamental symmetries.
