This is a first course on weak solutions of partial differential equations. The course begins with a brief introduction to distributions and weak derivatives. Next we consider Sobolev spaces and fundamental results: extension and trace theorems, Sobolev and Morrey theorem, Poincare's inequality and Rellich-Kondrachov theorem. Then we examine weak solutions of elliptic equations through Lax-Milgram theorem. The course ends with a discussions of weak solutions of linear evolution equations - second-order linear parabolic equations, linear hyperbolic systems and semigroup methods.