This contribution reviews the contents of a lecture given by M. Giaquinta in September 1992. It centers around the concept of area of a surface as well as the related questions concerning representation formulas for the area or the change of variable formula. The point of view is that of geometric measure theory of weak maps (as Sobolev maps) or of a parametric or non-parametric approach. Some open questions are also discussed. The presentation is along the following lines: 1) Sobolev and approximately differentiable functions (theorems of Calderón-Zygmund, Morrey-Sobolev, Kirszbraun, Rademacher and Federer); 2) the area formula (theorems of Rado-Reichelderfer); 3) the area of graphs and approximation in area.