Singular integral operators play a central role in modern harmonic analysis. Simplest examples of singular kernels are given by CalderónZygmund kernels. Many important properties of singular integrals have been thoroughly studied for CalderónZygmund operators. In the 1980's and early 1990's, Coifman, Weiss, and Christ noticed that the theory of CalderónZygmund operators can be generalized from Euclidean spaces to spaces of homogeneous type. The purpose of this book is to make the reader believe that homogeneity (previously considered as a cornerstone of the theory) is not needed. This claim is illustrated by presenting two harmonic analysis problems famous for their difficulty. The first problem treats semiadditivity of analytic and Lipschitz harmonic capacities. The volume presents the first selfcontained and unified proof of the semiadditivity of these capacities. The book details Tolsa's solution of Painlevé's and Vitushkin's problems and explains why these are problems of the theory of CalderónZygmund operators on nonhomogeneous spaces. The exposition is not dimensionspecific, which allows the author to treat Lipschitz harmonic capacity and analytic capacity at the same time. The second problem considered in the volume is a twoweight estimate for the Hilbert transform. This problem recently found important applications in operator theory, where it is intimately related to spectral theory of small perturbations of unitary operators. The book presents a technique that can be helpful in overcoming rather bad degeneracies (i.e., exponential growth or decay) of underlying measure (volume) on the space where the singular integral operator is considered. These situations occur, for example, in boundary value problems for elliptic PDE's in domains with extremely singular boundaries. Another example involves harmonic analysis on the boundaries of pseudoconvex domains that goes beyond the scope of CarnotCarathéodory spaces. The book is suitable for graduate students and research mathematicians interested in harmonic analysis. Readership Graduate students and research mathematicians interested in harmonic analysis. Reviews "...this book will interest anyone who would like to learn these new beautiful techniques in harmonic analysis and apply them..."  Hervé Pajot for Mathematical Reviews Table of Contents  Introduction
 Preliminaries on capacities
 Localization of Newton and Riesz potentials
 From distribution to measure. Carleson property
 Potential neighborhood that has properties (3.13)(3.14)
 The tree of the proof
 The first reduction to nonhomogeneous \(Tb\) theorem
 The second reduction
 The third reduction
 The fourth reduction
 The proof of nonhomogeneous Cotlar's lemma. Arbitrary measure
 Starting the proof of nonhomogeneous nonaccretive \(Tb\) theorem
 Next step in theorem 10.6. Good and bad functions
 Estimate of the diagonal sum. Remainder in theorem 3.3
 Twoweight estimate for the Hilbert transform. Preliminaries
 Necessity in the main theorem
 Twoweight Hilbert transform. Towards the main theorem
 Long range interaction
 The rest of the long range interaction
 The short range interaction
 Difficult terms and several paraproducts
 Twoweight Hilbert transform and maximal operator
 Bibliography
