It was established by Almgren at the beginning of the eighties that area-minimizing $n$-dimensional currents in Riemannian manifolds are regular up to a singular set of dimension at most $n-2$. To reach this goal Almgren developed an entirely new regularity theory, which occupies a very large monograph, published posthumously.  This talk is based on a series of joint works with Emanuele Spadaro, where we give alternative proofs to all Almgren's main steps, resulting into a much more manageable approach to his entire theory.