Hamiltonian dynamics, originated in classical mechanics, lies at the root of symplectic geometry and provides a wealth of examples. One of the central topics is the study of periodic orbits, and over the years many tools have been developed to investigate them. Among the most interesting and historically important examples is the three-body problem. Despite centuries of research, however, a complete picture of its periodic orbits is still lacking, with only partial results available. In this talk, I will present the basic ideas and techniques of Hamiltonian dynamics and symplectic geometry including Floer theory, and introduce the three-body problem and related systems. I will also briefly describe some of my recent results related with the rotating Kepler problem and three-body problem.