We reveal the existence of a new type of surface electromagnetic waves supported by hyperbolic metasurfaces, described by a conductivity tensor with an indefinite signature. We demonstrate that the spectrum of the hyperbolic metasurface waves consists of two branches corresponding to hybrid TE-TM waves with the polarization that varies from linear to elliptic or circular depending on the wave frequency and propagation direction. We analyze the effect of losses of the surface waves and derive the corresponding analytical asymptotic expressions.
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