  Classical mechanical systems are modeled by a symplectic manifold
$(M,\omega)$, and their symmetries, encoded in the action of a Lie group $G$ on
$M$ by diffeomorphisms that preserves $\omega$. These actions, which are called
"symplectic", have been studied in the past forty years, following the works of
Atiyah, Delzant, Duistermaat, Guillemin, Heckman, Kostant, Souriau, and
Sternberg in the 1970s and 1980s on symplectic actions of compact abelian Lie
groups that are, in addition, of "Hamiltonian" type, i.e. they also satisfy
Hamilton's equations. Since then a number of connections with combinatorics,
finite dimensional integrable Hamiltonian systems, more general symplectic
actions, and topology, have flourished. In this paper we review classical and
recent results on Hamiltonian and non Hamiltonian symplectic group actions
roughly starting from the results of these authors. The paper also serves as a
quick introduction to the basics of symplectic geometry.
