The theory of integral currents, developed by Federer and Fleming in the 60s, gives a powerful framework to solve the Plateau's problem in every dimension and codimension and general ambient manifolds. The interior and boundary regularity theory for the codimension one case is rather well understood, thanks to the work of several mathematicians in the 60s, 70s and 80s. In codimension higher than one the phenomenon of branching causes instead very serious problems. A celebrated monograph of Almgren provided a far-reaching interior regularity theorem. However, his original (typewritten) manuscript was more than 1700 pages long. Four years ago, in a series of works with Emanuele Spadaro we have given a substantially shorter and simpler version of Almgren's theory, building upon large portions of his program. This has allowed us to go beyond Almgren's result in several directions.