Mathematics for physics and engineering is traditionally covered by textbooks on "Mathematical Physics" or "Applied Mathematics." This book differs from those on pure mathematics and differs from lexicographic collections of methods for solving specific problems. Instead it emphasizes the mathematical concepts underlying manifold physical phenomena. The readers will not only acquire knowledges required for actual applications but also acquire the minimum "mathematical skills" necessary to study physics. The text is coherent and self-contained, states and proves a large number of theorems, lemmas, and corollaries that are relevant to physics and other related sciences. Extensive details on mathematical manipulations are provided. Each chapter contains a number of examples and practical exercises. Such a large number of examples provides the balance between mathematical formalisms and their applications. Reflecting the current interests, several new topics in developing fields, such as the mathematical background of quantum information theory and topology for the knot theory, are included.