Harmonic analysis on symmetric spaces euclidean space the sphere and the poincaré upper half plane This unique text is an introduction to harmonic analysis on the simplest symmetric spaces namely Euclidean space the sphere and the Poincaré upper half plane. This book is intended for beginning graduate students in mathematics or researchers in physics or engineering. Written with an informal style the book places an emphasis on motivation concrete examples history and above all applications in mathematics statistics physics and engineering. Many corrections new topics and updates have been incorporated in this new edition. These include discussions of the work of P. Sarnak and others making progress on various conjectures on modular forms the work of T. Sunada Marie France Vignras Carolyn Gordon and others on Mark Kacs question Can you hear the shape of a drum Ramanujan graphs wavelets quasicrystals modular knots triangle and quaternion groups computations of Maass waveforms and finally the authors comparisons of continuous theory with the finite analogues. Topics featured throughout the text include inversion formulas for Fourier transforms central limit theorems Poissons summation formula and applications in crystallography and number theory applications of spherical harmonic analysis to the hydrogen atom the Radon transform non Euclidean geometry on the Poincaré upper half plane H or unit disc and applications to microwave engineering fundamental domains in H for discrete groups tessellations of H from such discrete group actions automorphic forms the Selberg trace formula and its applications in spectral theory as well as number theory.