Integral currents are a natural generalization of oriented surfaces. The concept was pioneered by De Giorgi for hypersurfaces of the Euclidean space, and extended by Federer and Fleming to any codimension, general Riemannian ambients, and different coefficient groups. These classical works of the fifties and sixties established a general existence theory for the problems of finding surfaces of least area spanning a given contour and that of representing a fixed homology class with area minimizers. On the other hand, at the onset there are famous topological obstructions to the existence of smooth solutions, but even in the absence of these obstructions, celebrated examples show that minimizers are in general not smooth.

Even though I will mention the classical results for the hypersurfaces case, in these lectures I will focus on the case of general codimension and report on recent progress on the structure of singularities and on the existence of smooth approximations.