This text is an introduction to harmonic analysis on symmetric spaces, focusing on advanced topics such as higher rank spaces, positive definite matrix space and generalizations. This text explores the geometry and analysis of higher rank analogues of the symmetric spaces introduced in volume one. To illuminate both the parallels and differences of the higher rank theory, the space of positive matrices is treated in a manner mirroring that of the upper-half space in volume one. This concrete example furnishes motivation for the general theory of noncompact symmetric spaces, which is outlined in the final chapter. The book emphasizes motivation and comprehensibility, concrete examples and explicit computations (by pen and paper, and by computer), history, and, above all, applications in mathematics, statistics, physics, and engineering.
The second edition includes new sections on Donald St. P. Richards's central limit theorem for O(n)-invariant random variables on the symmetric space of GL(n, R), on random  matrix theory, and on advances in the theory of automorphic forms on arithmetic groups.

"Advanced graduate students and beginners in harmonic analysis on symmetric spaces are surely motivated and attracted by abundant examples, relevant history, and exercises. Excellent references in each section are useful for experts." (Takeshi Kawazoe, Mathematical Reviews, August, 2017)

"It is very nice to have available, now, the second, updated version of the entire set ? . Audrey Terras has done the mathematical community (and not just number theorists and modular formers) a great service: these books are a major contribution on several fronts, including the pedagogical one. They are of course also excellent references for various mathematical themes that are otherwise scattered all through the recent literature." (Michael Berg, MAA Reviews, maa.org, July, 2016)