Designed as an introduction to harmonic analysis and group representations, this book covers a wide range of topics rather than delving deeply into any particular one. In the words of H. Weyl "...it is primarily meant for the humble, who want to learn as new the things set forth therein, rather than for the proud and learned who are already familiar with the subject and merely look for quick and exact information..." . The main objective is to introduce the reader to concepts, ideas, results and techniques that evolve around symmetry-groups, representations and Laplacians . More specifically, the main interest concerns geometrical objects and structures { X }, discrete or continuous, that possess sufficiently large symmetry group G , such as regular graphs (Platonic solids), lattices, and symmetric Riemannian manifolds. All such objects have a natural Laplacian Delta , a linear operator on functions over X , invariant under the group action. There are many problems associated with Laplacians on X , such as continuous or discrete-time evolutions, on X , random walks, diffusion processes, and wave-propagation. This book contains sufficient material for a 1 or 2-semester course.