The topic of this monograph ranges beyond its title. We are in presence of a volume which gives a large survey of calculus of variations in several variables with detours into partial differential equations of variational type, but also captures some parts of mathematical programming and convex analysis. It is destined for graduate and postgraduate students, but more advanced students in mathematics will also find useful information in these pages, along with other scientists interested in applied mathematics. A clear presentation of the historical Dirichlet problem in Chapter 2 kicks off the first part. The classical tools of variational analysis such as weak topologies and the introduction of Sobolev spaces are introduced in this chapter. The Lax–Milgram theorem and the direct method in the calculus of variations are developed in Chapter 3 along with an introduction to convex optimization. An original and intuitive proof is given for the Ekeland variational principle based on the convergence of a discrete dynamical system. Chapter 4 contains complements on measure theory: Hausdorff measures, duality and introduction to Young measures, which makes the link between classical and modern calculus of variations. Chapter 5 deals with a complete survey of Sobolev spaces with some useful complements on capacity and potential theory. The next chapter contains classical applications to boundary value problems of the Dirichlet and Neumann type, along with an introduction to the p-Laplacian.