Some model problems: the Brachistochrone problem, the Fermat principle, minimal surfaces of revolution, the Dirichlet functional, minimal surfaces. The Classical Method. The fundamental lemma of the Calculus of Variations. Necessary conditions for minimality: the Euler-Lagrange equation. Second form of the Euler-Lagrange equation. Hamiltonian formulation. Fields theory. An introduction to Sobolev Spaces (definition and main properties. Examples. Embeddings. Duals and Weak convergence. Rellich-Kondrachov Theorem. Poincare' Inequalities). Direct Methods. Sufficient conditions for weak lower semicontinuity in Sobolev spaces. Necessary conditions for weak lower semicontinuity in Sobolev spaces: the scalar and the vectorial case. Polyconvexity. Relaxation theory. Regularity of minimisers. Minimal surfaces. The Isoperimetric Inequality.