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 we will discuss existence and uniqueness theorems for such surfaces, focusing on a uniqueness result for free boundary minimal annuli. this is joint work with m. li and r. schoen.