  Non-uniform metasurfaces (electrically thin composite layers) can be used for
shaping refracted and reflected electromagnetic waves. However, known design
approaches based on the generalized refraction and reflection laws do not allow
realization of perfectly performing devices: there are always some parasitic
reflections into undesired directions. In this paper we introduce and discuss a
general approach to the synthesis of metasurfaces for full control of
transmitted and reflected plane waves and show that perfect performance can be
realized. The method is based on the use of an equivalent impedance matrix
model which connects the tangential field components at the two sides on the
metasurface. With this approach we are able to understand what physical
properties of the metasurface are needed in order to perfectly realize the
desired response. Furthermore, we determine the required polarizabilities of
the metasurface unit cells and discuss suitable cell structures. It appears
that only spatially dispersive metasurfaces allow realization of perfect
refraction and reflection of incident plane waves into arbitrary directions. In
particular, ideal refraction is possible only if the metasurface is
bianisotropic (weak spatial dispersion), and ideal reflection without
polarization transformation requires spatial dispersion with a specific,
strongly non-local response to the fields.
