This is the first of two volumes (the second will deal with applications) and it is focused on the interplay between pure and applied nonlinear analysis. The interest for this field in the last few decades has been huge, mainly because nonlinear analysis is the outgrowth of the needs of concrete problems arising in the applied sciences. In this volume the emphasis is on those basic abstract methods and theories that are useful in the study of nonlinear boundary value problems. The next volume 2 will be devoted to the study of such problems. The contents of the first volume are divided into six chapters, plus a comprehensive list of references and subject index. This work provides researchers and graduate students with a thorough introduction to abstract methods used in the variational and topological analysis of nonlinear boundary value problems described by stationary differential operators. We give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations, as well as their applications to various processes arising in the applied sciences. We demonstrate how all of these diverse topics are connected to other important parts of mathematics, including topology, functional analysis, mathematical physics, and potential theory. Throughout the book we maintain a good balance between rigorous mathematics and physical applications. The main benefits of this book are: (i) it is a bridge among several fields of mathematics; (ii) it illustrates the power of Sobolev spaces in nonlinear analysis; (iii) the methods are developed in their generality and can be extended to other classes of nonlinear problems; and (iv) it uses a modern, unified approach to analyzing nonlinear boundary value problems. The primary audience for this book are researchers in pure and applied nonlinear analysis and graduate students.