  Floer invented his theory in the mid eighties in order to prove the Arnol'd
conjectures on the number of fixed point of Hamiltonian diffeomorphisms and
Lagrangian intersections. Over the last thirty years, many versions of Floer
homology have been constructed. In symplectic and contact dynamics and geometry
they have become a principal tool, with applications that go far beyond the
Arnol'd conjectures: The proof of the Conley conjecture and of many instances
of the Weinstein conjecture, rigidity results on Lagrangian submanifolds and on
the group of symplectomorphisms, lower bounds for the topological entropy of
Reeb flows and obstructions to symplectic embeddings are just some of the
applications of Floer's seminal ideas. Other Floer homologies are of
topological nature. Among their applications are Property P for knots and the
construction of compact topological manifolds of dimension greater than five
that are not triangulisable.
  This is by no means a comprehensive survey on the presently known Floer
homologies and their applications. Such a survey would take several hundred
pages. We just describe some of the most classical versions and applications,
together with the results that we know or like best. The text is written for
non-specialists, and the focus is on ideas rather than generality. Two
intermediate sections recall basic notions and concepts from symplectic
dynamics and geometry.
