This book is written for both physicists and mathematicians. The topics treated include Newtonian mechanics, semi-classical mechanics, (non-relativistic) quantum mechanics and its Bohmian interpretation. The main tool in the book is symplectic geometry. A study of symplectic rigidity leads to a semi-classical quantization scheme and to the Maslov index. A use of a general Leray index leads to a definition of a wave form on the phase space. The metaplectic group is a double cover of the symplectic group. A study of its representations is used in a treatment of the Schrödinger equation for a class of Hamiltonians and for a definition of certain Feynman path integrals.