In Chaps. 4 and 6 , we obtained existence of weak solutions to quite general elliptic equations, through some minimization problems. This chapter is devoted to inquire to which extent such weak solutions are actually solutions in a classical sense. This is the goal of the Regularity Theory : prove that a function is actually more regular than just merely Sobolev or Lipschitz, thanks to the fact that it minimizes a functional or weakly solve a partial differential equation. We will avoid general statements and stick to the very simple, yet meaningful, example of weakly harmonic functions. The chapter use this model problem to neatly illustrate De Giorgi-Moser-type techniques to gain regularity. A list of problems will then guide the reader through some (possibly nonlinear) generalizations.