In this course, we study various existence and regularity theories for elliptic partial differential equations and systems, though the emphasis is on regularity. I plan to treat the Leray-Schauder theory, the existence and regularity of critical points of elliptic integrands (scalar and vectorial), the solution of the XIX Hilbert problem (De Giorgi's proof and Moser's iteration), the theory of viscosity solutions of fully nonlinear elliptic equations.