"Let be a smooth Riemannian manifold, a smooth closed oriented submanifold of codimension higher than 2 and T an integral area-minimizing current in which bounds . We prove that the set of regular points of T at the boundary is dense in . Prior to our theorem the existence of any regular point was not known, except for some special choice of and . As a corollary of our theorem we answer to a question in Almgren's Almgren's big regularity paper from 2000 showing that, if is connected, then T has at least one point p of multiplicity 1 2 , namely there is a neighborhood of the point p where T is a classical submanifold with boundary ; we generalize Almgren's connectivity theorem showing that the support of T is always connected if is connected; we conclude a structural result on T when consists of more than one connected component, generalizing a previous theorem proved by Hardt and Simon in 1979 when = Rm+1 and T is m-dimensional"--

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