<p>The three-body problem continues to be an unsolved challenge in the scientific community. It serves as a foundational model for exploring nonlinear dynamics, astrophysics, and orbital mechanics. Due to the inherently chaotic nature of the three-body problem, periodic orbits are regarded as the primary framework that forms the backbone of the system, which provides crucial insights into its dynamical properties. In this review, we examine both historical and recent advances in the study of periodic orbits for the planar three-body problem. We provide an overview of the general three-body problem, the definition and identification of numerical periodic orbits, and the topological classification methods used to categorize these solutions. Additionally, we describe the clean numerical simulation (CNS) employed to obtain accurate trajectories, as well as both analytical and numerical results for periodic orbits in systems with different initial configurations, with equal or unequal masses, with or without angular momentum. The review also covers key topics such as linear stability analysis and the extension of Kepler’s third law to general three-body problems. Finally, we highlight several open questions and promising directions for future research in this field.</p>