This book presents a modern account of how microlocal methods can be used to solve problems in integral geometry and tomography. Focusing on the X-ray, Radon, geodesic X-ray, light ray, and related transforms, it explains how one can recover functions and tensor fields, and particularly their singularities, from integrals over curves or hyperplanes. Beginning with Euclidean models and progressing to geodesic, weighted, attenuated, magnetic, dynamic, and generalized transforms, the book connects classical Euclidean inversion formulas with the analysis of more general transforms, using the calculus of wave front sets, pseudodifferential operators, Fourier integral operators, and analytic microlocal analysis. The book focuses on uniqueness, stability, Fredholm properties, and generic uniqueness, including situations with limited data, conjugate points, tensorial gauge freedoms, and failure of local microlocal invertibility. Applications and motivations range from CT and SAR imaging to seismology, cosmology, boundary rigidity, and inverse problems in Riemannian geometry. Written for graduate students and researchers in analysis, geometry, and inverse problems, it offers a detailed guide to the microlocal structure behind modern integral geometry. The concise appendices cover a broad range of background material and technical tools, helping make the book more self-contained without interrupting the main exposition.
