Free boundary minimal surfaces in the ball are proper branched minimal
immersions of a surface into the ball that meet the boundary of the ball
orthogonally. Such surfaces have been extensively studied, and they arise as
extremals of the area functional for relative cycles in the ball. They also
arise as extremals of an eigenvalue problem on surfaces with boundary. In
this talk I will describe uniqueness (joint work with R. Schoen) and
compactness (joint work with M. Li) theorems for such surfaces.