Symplectic structures underlie the equations of classical mechanics, and their properties are reflected in the behavior of a wide range of physical systems. Over the years much detailed information has accumulated about the behavior of particular systems. Powerful new methods, such as Gromov's flexibility theorem and proofs of the Arnold conjectures, have produced striking results, but the modern global theory of symplectic topology has only recently emerged. This book is an introduction to the subject for postgraduate students, presenting new methods in the field and providing proofs of the simpler versions of the most important new theorems. The deepest theorems in the book are proved by a new finite dimensional variational analysis which combines ideas from Viterbo's generating function approach with the infinite dimensional variational analysis of Hofer-Zehnder. Exercises are also included.