  We develop a parametric cut finite element method for elliptic boundary value
problems with corner singularities where we have weighted control of higher
order derivatives of the solution to a neighborhood of a point at the boundary.
Our approach is based on identification of a suitable mapping that grades the
mesh towards the singularity. In particular, this mapping may be chosen without
identifying the opening angle at the corner. We employ cut finite elements
together with Nitsche boundary conditions and stabilization in the vicinity of
the boundary. We prove that the method is stable and convergent of optimal
order in the energy norm and $L^2$ norm. This is achieved by mapping to the
reference domain where we employ a structured mesh.
