1 Introduction.- 1.1 Symplectic vector spaces.- 1.2 Symplectic diffeomorphisms and Hamiltonian vector fields.- 1.3 Hamiltonian vector fields and symplectic manifolds.- 1.4 Periodic orbits on energy surfaces.- 1.5 Existence of a periodic orbit on a convex energy surface.- 1.6 The problem of symplectic embeddings.- 1.7 Symplectic classification of positive definite quadratic forms.- 1.8 The orbit structure near an equilibrium, Birkhoff normal form.- 2 Symplectic capacities.- 2.1 Definition and application to embeddings.- 2.2 Rigidity of symplectic diffeomorphisms.- 3 Existence of a capacity.- 3.1 Definition of the capacity c0.- 3.2 The minimax idea.- 3.3 The analytical setting.- 3.4 The existence of a critical point.- 3.5 Examples and illustrations.- 4 Existence of closed characteristics.- 4.1 Periodic solutions on energy surfaces.- 4.2 The characteristic line bundle of a hypersurface.- 4.3 Hypersurfaces of contact type, the Weinstein conjecture.- 4.4 “Classical” Hamiltonian systems.- 4.5 The torus and Herman’s Non-Closing Lemma.- 5 Compactly supported symplectic mappings in ?2n.- 5.1 A special metric d for a group D of Hamiltonian diffeomorphisms.- 5.2 The action spectrum of a Hamiltonian map.- 5.3 A “universal” variational principle.- 5.4 A continuous section of the action spectrum bundle.- 5.5 An inequality between the displacement energy and the capacity.- 5.6 Comparison of the metric d on D with the C0-metric.- 5.7 Fixed points and geodesics on D.- 6 The Arnold conjecture, Floer homology and symplectic homology.- 6.1 The Arnold conjecture on symplectic fixed points.- 6.2 The model case of the torus.- 6.3 Gradient-like flows on compact spaces.- 6.4 Elliptic methods and symplectic fixed points.- 6.5 Floer’s appraoch to Morse theory for the actionfunctional.- 6.6 Symplectic homology.- A.2 Action-angle coordinates, the Theorem of Arnold and Jost.- A.4 The Cauchy-Riemann operator on the sphere.- A.5 Elliptic estimates near the boundary and an application.- A.6 The generalized similarity principle.- A.7 The Brouwer degree.- A.8 Continuity property of the Alexander-Spanier cohomology.