Closed geodesic on M, means a geodesic R->M which is periodic. It's not hard do see, in any compact manifold not simply-connected, there is a closed geodesic. Also, in every simply-connected one, there is a closed geodesic, which is proved using a "Birkhoff curve shortening process". People try to bound the length of the shortest closed geodesic, by diameter and volume of M. I will focus on recent works on S^2, so the whole process is visible to the audience. Reference: 1.[Croke 1988] Area and the length of the shortest closed geodesic 2.[Nabutovsky and Rotman 2004] Volume, diameter and the minimal mass of a stationary 1-cycle 3.[Nabutovsky and Rotman 2002] The length of the shortest closed geodesic on a 2-dimensional sphere 4.[Sabourau 2004] Filling radius and short closed geodesic of the two-sphere Those are online at www.math.upenn.edu/~zx/Croke1988.pdf www.math.upenn.edu/~zx/Nabutovsky-Rotman_1-cycle.pdf www.math.upenn.edu/~zx/Nabutovsky-Rotman_4D.pdf www.math.upenn.edu/~zx/Sabourau4D.pdf