Starting from undergraduate level, this book systematically develops the basics of - Analysis on manifolds, Lie groups and G-manifolds (including equivariant dynamics) - Symplectic algebra and geometry, Hamiltonian systems, symmetries and reduction, - Integrable systems, Hamilton-Jacobi theory (including Morse families, the Maslov class and caustics). The first item is relevant for virtually all areas of mathematical physics, while the second item provides the basis of Hamiltonian mechanics. The last item introduces to important special areas. Necessary background knowledge on topology is provided in an appendix. The aim of this book is to enable the reader to access research monographs on more advanced topics. The style of this book is that of a mathematics textbook, with full proofs given in the text or as exercises. All material is illustrated by detailed examples, a number of which is taken up repeatedly for demonstrating how the methods evolve and interact.