1. The Laplace equation as the prototype of an elliptic partial differential equation of second order -- 2. The maximum principle -- 3. Existence techniques I: methods based on the maximum principle -- 4. Existence techniques II: parabolic methods. The heat equation -- 5. The wave equation and its connections with the Laplace nad heat equations -- 6. The heat equation, semigroups, and Brownian motion -- 7. The Dirichlet principle. Variational methods for the solution of PDEs (existence techniques III) -- 8. Sobolev spaces and L[superscript 2] regularity theory -- 9. Strong solutions -- 10. The regularity theory of Schauder and the continuity method (existence techniques IV) -- 11. The Moser iteration method and the regularity theorem of de Giorgi and Nash -- Appendix: Banach and Hilbert spaces. The L[superscript p]-spaces.