This textbook introduces to the vector and tensor calculus as it is used in continuum mechanics. Therefore, it limits itself to the 3-dimensional Euclidian space. It contains, in addition, an introduction to continuum mechanics and linear shell theory. The theoretical concepts of small and finite strain elasticity, its numerical formulation and essential implementation aspects are presented in detail. After setting the variational framework of boundary value problems, several Finite Element representations are discussed. The formulation of isotropic and anisotropic elasticity is performed within a basis-independent formalism. As far as possible illustrative approaches are chosen although some of the concepts require axiomatic introduction. All chapters widely consist of examples and exercises. This book is written mainly for students in civil and mechanical engineering and bridges the gap between mathematics and its engineering application.