An oriented hypersurface in a hyperkaehler 4-manifold naturally inherits a coclosed coframing.  Bryant showed that, in the real analytic case, any oriented 3-manifold with a coclosed coframing can always be locally “thickened” to a hyperkaehler 4-manifold, in an essentially unique way.  This raises the natural question: when can these 3-manifolds with this structure arise as the boundary of a hyperkaehler 4-manifold?  In particular, starting from a compact hyperkaehler 4-manifold with boundary, which deformations of the boundary structure can be extended to a hyperkaehler deformation of the interior?  I will discuss recent progress on this problem, which is joint work with Joel Fine and Michael Singer.