This book provides a thorough and selfcontained introduction to the \(\bar{\partial}\)Neumann problem, leading up to current research, in the context of the \(\mathcal{L}^{2}\)Sobolev theory on bounded pseudoconvex domains in \(\mathbb{C}^{n}\). It grew out of courses for advanced graduate students and young researchers given by the author at the Erwin Schrödinger International Institute for Mathematical Physics and at Texas A & M University. The introductory chapter provides an overview of the contents and puts them in historical perspective. The second chapter presents the basic \(\mathcal{L}^{2}\)theory. Following is a chapter on the subelliptic estimates on strictly pseudoconvex domains. The two final chapters on compactness and on regularity in Sobolev spaces bring the reader to the frontiers of research. Prerequisites are a solid background in basic complex and functional analysis, including the elementary \(\mathcal{L}^{2}\)Sobolev theory and distributions. Some knowledge in several complex variables is helpful. Concerning partial differential equations, not much is assumed. The elliptic regularity of the Dirichlet problem for the Laplacian is quoted a few times, but the ellipticity results needed for elliptic regularization in the third chapter are proved from scratch. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. Readership Graduate students and research mathematicians interested in \(\mathcal{L}^{2}\)Sobolev Theory. Table of Contents  Introduction
 The \(\mathcal{L}^2\)theory
 Strictly pseudoconvex domains
 Compactness
 Regularity in Sobolev spaces
 Bibliography
 Index
