|The discoveries of the last decade have shed new light on the old field of Hamiltonian systems and have led to the creation of a new field: symplectic topology. Surprising rigidity phenomena demonstrate that the nature of symplectic mappings is very different from that of volume preserving mappings. This raises new questions, many of them still unanswered. On the other hand, analysis of an old variational principle in classical mechanics has established global periodic phenomena in Hamiltonian systems. As it turns out, these seemingly different phenomena are mysteriously related. One of the links is a class of symplectic invariants, called symplectic capacities. These invariants are the main theme of this book, which includes such topics as basic symplectic geometry, symplectic capacities and rigidity, periodic orbits for Hamiltonian systems and the least action principle, a bi-invariant metric on the symplectic diffeomorphism group and its geometry, symplectic fixed point theory and first order elliptic systems, the Arnold conjectures and a survey on Floer- and symplectic homology. The exposition is self-contained and addressed to researchers and students from the graduate level onwards.