Dark matter structures within strong gravitational lens galaxies and along their lines of sight leave a gravitational imprint on the multiple images of lensed sources. Strong gravitational lensing provides, therefore, a key test of different dark matter models. In this article, we describe how galaxy-scale strong gravitational lensing observations are sensitive to the physical nature of dark matter. We provide an historical perspective of the field, and review its current status. We discuss the challenges and advances in terms of data, treatment of systematic errors and theoretical predictions, that will enable one to deliver a stringent and robust test of different dark matter models in the next decade. With the advent of the next generation of sky surveys, the number of known strong gravitational lens systems is expected to increase by several orders of magnitude. Coupled with high-resolution follow-up observations, these data will provide a key opportunity to constrain the properties of dark matter with strong gravitational lensing.The Max Planck Society for support through a Max Planck Lise Meitner Group and funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme; the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodovska-Curie grant agreement No 897124; a Gliese Fellowship; the National Science Foundation; a HQP grant from the McDonald Institute; the Schmidt Futures Foundation; the National Sciences and Engineering Council of Canada; the Fonds de recherche du Québec; the Canada Research Chairs Program; the Netherlands Organization for Scientific Research; the Chinese Academy of Sciences and the National Research Foundation of South Africa. Open Access funding enabled and organized by Projekt DEAL.http://link.springer.com/journal/11214hj2024PhysicsNon
