(Boundary) Regularity for area minimizing surfaces

Plateau problem consists in finding a surface of minimal area among the ones spanning a given curve. It is among the oldest problem in the calculus of variations and its study lead to wonderful development in mathematics.
Federer and Fleming integral currents provide a suitably weak solution to the Plateau problem in arbitrary Riemannian manifolds, in any dimension and
co-dimension. Once this week solution has been found a natural question consists in understanding whether it is classical one. i.e. a smooth minimal surface. This is the topic of the regularity theory, which naturally splits into interior regularity and boundary regularity.
After the monumental work of Almgren, revised by De Lellis and Spadaro, interior regularity is by now well understood. Boundary regularity is instead less clear and some new phenomena appear.
Aim of the talk is to give an overview of the problem and to present some boundary regularity results we have obtained in the last years.