This book derives from author Nolan R. Wallach's notes for a course on symplectic geometry and Fourier analysis, which he delivered at Rutgers University in 1975 for an audience of graduate students in mathematics and their professors. The monograph is geared toward readers who have taken a basic course in differential manifolds and elementary functional analysis. The first chapters cover certain geometric preliminaries, advancing to discussions of symplectic geometry and the application of its concepts to the action of a Lie group on a symplectic manifold. Subsequent chapters address Fourier analysis, the metaplectic representation, and quantization. A final chapter on the Kirillov theory applies the ideas of the previous chapters to homogeneous symplectic manifolds of nilpotent Lie groups. The book concludes with an Appendix on Quantum Mechanics by Robert Hermann.