  Maxwell's boundary conditions (MBCs) were long known insufficient to
determine the interaction of light with spatially dispersive medium.
Supplementing MBCs with additional boundary conditions (ABCs) has become a
normal yet controversial practice. Here the problem is solved by carefully
analyzing some subtle aspects of a physical surface. A generic formalism based
on a non-local susceptibility is presented for handling the interaction of
light with the surfaces of an arbitrary medium and applied to study the
traditional problem of exciton polaritons. We show that ABCs can always be
contrived to fit the theory but they can by no means be construed as intrinsic
surface characteristics, which are instead captured by a \textit{surface
response function}. Methods for experimentally extracting the spatial profile
of this function are proposed.
