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This is a survey paper on Morse theory and the existence problem for closedgeodesics. The free loop space plays a central role, since closed geodesics arecritical points of the energy functional. As such, they can be analyzed throughvariational methods. The topics that we discuss include: Riemannian background,the Lyusternik-Fet theorem, the Lyusternik-Schnirelmann principle ofsubordinated classes, the Gromoll-Meyer theorem, Bott's iteration of the indexformulas, homological computations using Morse theory, $SO(2)$- vs.$O(2)$-symmetries, Katok's examples and Finsler metrics, relations tosymplectic geometry, and a guide to the literature.  The Appendix written by Umberto Hryniewicz gives an account of the problem ofthe existence of infinitely many closed geodesics on the $2$-sphere.

Morse theory, closed geodesics, and the homology of free loop spaces