Given a curve $\Gamma$, what is the surface $T$ that has least area among all surfaces spanning $\Gamma$? This classical problem and its generalizations are called Plateau's problem. In this talk we consider area minimizers among the class of integral currents, or roughly speaking, orientable submanifolds. Since the 1960s a lot of work has been done by De Giorgi, Almgren, et al to study the regularity of these minimizers at the interior. Much less is known about regularity at the boundary (in the case of codimension greater than 1). Recently, De Lellis et al. have found surprising examples of boundary singularity even when the prescribed curve $\Gamma$ is smooth. I will speak about some recent progress in this direction and my joint work with C. De Lellis.