  These notes combine material from short lecture courses given in Paris,
France, in July 2001 and in Srni, the Czech Republic, in January 2003. They
discuss groups of symplectomorphisms of closed symplectic manifolds (M,\om)
from various points of view. Lectures 1 and 2 provide an overview of our
current knowledge of their algebraic, geometric and homotopy theoretic
properties. Lecture 3 sketches the arguments used by Gromov, Abreu and
Abreu-McDuff to figure out the rational homotopy type of these groups in the
cases M= CP^2 and M=S^2\times S^2. We outline the needed J-holomorphic curve
techniques. Much of the recent progress in understanding the geometry and
topology of these groups has come from studying the properties of fibrations
with the manifold M as fiber and structural group equal either to the
symplectic group or to its Hamiltonian subgroup Ham(M). The case when the base
is S^2 has proved particularly important. Lecture 4 describes the geometry of
Hamiltonian fibrations over S^2, while Lecture 5 discusses their Gromov-Witten
invariants via the Seidel representation. It ends by sketching Entov's
explanation of the ABW inequalities for eigenvalues of products of special
unitary matrices. Finally in Lecture 6 we apply the ideas developed in the
previous two lectures to demonstrate the existence of (short) paths in
Ham(M,\om) that minimize the Hofer norm over all paths with the given
endpoints.
