  In this paper, we prove uniform curvature estimates for immersed stable free
boundary minimal hypersurfaces which satisfy a uniform area bound. Our result
is a natural generalization of the celebrated Schoen-Simon-Yau interior
curvature estimates up to the free boundary. A direct corollary of our
curvature estimates is a smooth compactness theorem which is an essential
ingredient in the min-max theory of free boundary minimal hypersurfaces
developed by the last two authors. We also prove a monotonicity formula for
free boundary minimal submanifolds in Riemannian manifolds for any dimension
and codimension. For the case of $3$-manifolds with boundary, we prove a
stronger curvature estimate for properly embedded stable free boundary minimal
surfaces without any assumption on the area bound. This generalizes Schoen's
interior curvature estimates to the free boundary setting. Our proof uses the
theory of minimal laminations developed by Colding and Minicozzi.
