The problem of closed geodesics is a traditional and
important topic in dynamical systems and differential geometry.  There is a longstanding conjecture that there exist infinitely many distinct closed geodesics on every compact Riemannian manifold. The current interest on this problem is on compact simply connected manifolds, specially spheres. So far not much is known on the multiplicity of closed geodesics on such manifolds when their dimensions are at least 3. Recently, Professor Yiming Long and myself have proved the following

Theorem: There exist at least two distinct closed geodesics on every compact simply connected Riemannian (or Finsler) manifold whose dimension is at least 2.

In this talk, I shall give a survey on the study of the problem of closed geodesics and explain some main ideas in the proof of the above theorem.