  In this paper, we consider locally wedge-shaped manifolds, which are
Riemannian manifolds that are allowed to have both boundary and certain types
of edges. We define and study the properties of free boundary minimal
hypersurfaces inside locally wedge-shaped manifolds. In particular, we show a
compactness theorem for free boundary minimal hypersurfaces with curvature and
area bounds in a locally wedge-shaped manifold. Additionally, using
Schoen-Simon-Yau's estimates, we also prove a Bernstein-type theorem indicating
that, under certain conditions, a stable free boundary minimal hypersurface
inside a Euclidean wedge must be a portion of a hyperplane. As our main
application, we establish a curvature estimate for sufficiently regular free
boundary minimal hypersurfaces in a locally wedge-shaped manifold with certain
wedge angle assumptions. We expect this curvature estimate will be useful for
establishing a min-max theory for the area functional in wedge-shaped spaces.
