Interactions between quantum particles, such as electrons, are the source of important effects, ranging from superconductivity, to the formation of molecular bonds, or the stability of elementary compounds at high-energies. In this article, we illustrate how advances in the cold-atom community to further engineer such long-range interactions have stimulated the simulation of new regimes of these fundamental many-body problems. The goal is two-fold: first, to provide a comprehensive review of the different strategies proposed and/or experimentally realized to induce long-range interactions among atoms moving in optical potentials. Second, to showcase various fields where such platforms can offer new insights, ranging from the simulation of condensed matter phenomena to the study of lattice gauge theories, and the simulation of electronic configurations in chemistry. We then discuss the challenges and opportunities of these platforms compared to other complementary approaches based on digital simulation and quantum computation.