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