«From reviews of the original edition:

""The book is excellent both as a text and as a reference work; it will clearly become another instant classic."" —American Scientist

""This volume makes an excellent companion to the author's Differential Geometry, Lie Groups, and Symmetric Spaces, putting to work many of the abstract concepts developed in the earlier volume. The introductory material and large number of exercises (with answers!) will make the book quite appropriate for students. Researchers will find numerous useful references on geometric analysis, along with proofs, connections with other parts of mathematics, and valuable historical remarks. This book, like the author's previous work on differential geometry, will no doubt inspire considerable further research and become the standard text on the subjects it covers."" —Mathematical Reviews

""Few treatises today can lay claim to being ""aere perennius"", but all of Helgason's books certainly do with a vengeance ... [He] sets a model of style and clarity that has not been matched since Enriques's Geometria proiettiva. This is the kind of mathematics that will live forever."" —The Bulletin of Mathematics Books

""A most valuable contribution to Lie theory and to the interplay between geometry and analysis. It is remarkable that the beautiful theory in Chapter IV can be presented in a textbook form with complete proofs."" —Bulletin of the London Mathematical Society

""The diversity of subjects treated is great. Nevertheless the author has managed to achieve coherence of presentation by clearly putting forward a few main themes and basic problems. The first third of the book is suitable as a text for beginning graduate students; the book is also an excellent source of reference for experts. No doubt it will become a new standard in the field."" —CWI Quarterly»