<p>This article presents a contemporary survey of mathematical advances in Choquard-type equations, a class of nonlinear, nonlocal partial differential equations with profound applications in mathematical physics and analysis. The survey synthesizes foundational results and recent progress regarding existence, regularity, and qualitative properties of solutions, including ground states, multiplicity, and topological effects in bounded domains. It highlights developments for equations with fractional Laplacians, singular and doubly nonlocal nonlinearities, and Kirchhoff-type operators. Alongside technical advances, the article outlines open research problems and underscores both the analytical challenges and rich structures inherent to nonlocal elliptic equations, serving as a comprehensive resource for researchers in nonlinear analysis and mathematical physics.</p>