The provided text is the preface and table of contents for the third edition of "The Mathematical Theory of Finite Element Methods" by Susanne C. Brenner and L. Ridgway Scott, published by Springer. The book is part of the "Texts in Applied Mathematics" series, edited by J.E. Marsden, L. Sirovich, and S.S. Antman. The series aims to meet the current and future needs of advances in applied mathematics and to encourage the teaching of new courses. The preface to the third edition highlights four new sections: the BDDC domain decomposition preconditioner, a convergent adaptive algorithm, interior penalty methods, and Poincaré-Friedrichs inequalities for piecewise $W_p^1$ functions. The authors have also made improvements throughout the text, added new exercises, and updated the list of references. They acknowledge the support from the National Science Foundation and the Alexander von Humboldt Foundation. The preface to the second edition mentions the addition of two new chapters: one on additive Schwarz theory and multilevel domain decomposition preconditioners, and another on *a posteriori* error estimators and adaptivity. The authors thank various individuals and institutions for their contributions and support. The preface to the first edition outlines the book's purpose, which is to develop the basic mathematical theory of the finite element method, a widely used technique in engineering design and analysis. It is intended for graduate students and mathematically sophisticated engineers and scientists. The book covers essential material from Chapters 0 through 5, with additional chapters providing more advanced topics. The authors describe three possible course paths based on the content. The contents of the book include chapters on basic concepts, Sobolev spaces, variational formulation of elliptic boundary value problems, construction of finite element spaces, polynomial approximation theory, n-dimensional variational problems, finite element multigrid methods, additive Schwarz preconditioners, max-norm estimates, adaptive meshes, variational crimes, applications to planar elasticity, mixed methods, iterative techniques for mixed methods, and applications of operator-interpolation theory. Each chapter includes exercises and references.The provided text is the preface and table of contents for the third edition of "The Mathematical Theory of Finite Element Methods" by Susanne C. Brenner and L. Ridgway Scott, published by Springer. The book is part of the "Texts in Applied Mathematics" series, edited by J.E. Marsden, L. Sirovich, and S.S. Antman. The series aims to meet the current and future needs of advances in applied mathematics and to encourage the teaching of new courses. The preface to the third edition highlights four new sections: the BDDC domain decomposition preconditioner, a convergent adaptive algorithm, interior penalty methods, and Poincaré-Friedrichs inequalities for piecewise $W_p^1$ functions. The authors have also made improvements throughout the text, added new exercises, and updated the list of references. They acknowledge the support from the National Science Foundation and the Alexander von Humboldt Foundation. The preface to the second edition mentions the addition of two new chapters: one on additive Schwarz theory and multilevel domain decomposition preconditioners, and another on *a posteriori* error estimators and adaptivity. The authors thank various individuals and institutions for their contributions and support. The preface to the first edition outlines the book's purpose, which is to develop the basic mathematical theory of the finite element method, a widely used technique in engineering design and analysis. It is intended for graduate students and mathematically sophisticated engineers and scientists. The book covers essential material from Chapters 0 through 5, with additional chapters providing more advanced topics. The authors describe three possible course paths based on the content. The contents of the book include chapters on basic concepts, Sobolev spaces, variational formulation of elliptic boundary value problems, construction of finite element spaces, polynomial approximation theory, n-dimensional variational problems, finite element multigrid methods, additive Schwarz preconditioners, max-norm estimates, adaptive meshes, variational crimes, applications to planar elasticity, mixed methods, iterative techniques for mixed methods, and applications of operator-interpolation theory. Each chapter includes exercises and references.