  The paper presents partial overview of the mathematical synthesis and the
physical realization of metasurfaces, and related illustrative examples. The
synthesis consists in determining the exact tensorial surface susceptibility
functions of the metasurface, based on generalized sheet transition conditions,
while the realization deals with both metallic and dielectric scattering
particle structures. The examples demonstrate the capabilities of the synthesis
and realization techniques, thereby showing the plethora of possible
metasurface field transmission and subsequent applications. The first example
is the design of two diffraction engineering birefringent metasurfaces
performing polarization beam splitting and orbital angular momentum
multiplexing, respectively. Next, we discuss the concept of the "transistor"
metasurface, which is an electromagnetic linear switch based on destructive
interferences. Then, we introduce a non-reciprocal non-gyrotropic metasurface
using a pick-up circuit radiator (PCR) architecture. Finally, the
implementation of all-dielectric metasurfaces for spatial dispersion
engineering is discussed.
