From the reviews:

“Beginning graduate students and interested nonspecialists will gain from the book because of its clear exposition and comprehensive nature. Practitioners of integral geometry will find it a valuable reference with complete and clear proofs as well as specialized items of interest … . This book is a well-written and beautiful introduction to integral geometry from the perspective of group actions. It has valuable thought provoking exercises. … It should be read by anyone who would like to learn more about integral geometry.” (Fulton Gonzalez and Eric Todd Quinto, Bulletin of the American Mathematical Society, Vol. 50 (4), October, 2013)

“This specialist book in the field of geometric analysis is intended for graduate students and professionals. … The field has several applications including medical X-ray technology and tomography. The nine-chapter volume devotes a chapter to these technologies and other applications. The book is largely self-contained, and includes a chapter at the end with necessary background material … . The volume includes a substantial bibliography, and each chapter concludes with bibliographic notes. Summing Up: Recommended. Graduate students and above.” (D. Z. Spicer, Choice, Vol. 48 (11), August, 2011)

“Integral Geometry and Radon Transforms is truly a pedagogical contribution by a master in this field, aiming at providing proper background – and then some – for entry into real work on the topics he brings into the spotlight. … every page is filled with serious mathematics, and Helgason provides a lot of commentary and references that ought to be pursued by those wishing to enter the field. It is a marvelous example of fine scholarship.” (Michael Berg, The Mathematical Association of America, January, 2011)

“The book under review is devoted to integral operators of geometric nature. The monograph deals with the problem of representing a function on a manifold in terms of its integrals over certain submanifolds – hence the term Radon transform. … The book will be of interest for both students and specialists in integral geometry, analysis on homogeneous spaces and partial differential equations. It may serve as a source for further research in the area.” (Valery Vladimirovich Volchkov, Zentralblatt MATH, Vol. 1210, 2011)