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We reveal the existence of a type of surface electromagnetic wave 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 transverse electric-transverse magnetic waves with a polarization that varies from linear to elliptic or circular depending on the wave frequency and propagation direction. The shape of the equal-frequency contours drastically depends on the frequency and changes from the elliptical to hyperbolic—a topological transition takes place. We derive asymptotic formulas describing the losses of the surface waves for capacitive, inductive, and hyperbolic regimes of the metasurface. We analyze numerically the generation of surface waves by a point electric dipole placed in the vicinity of the metasurface with numerical simulations, and also reveal wave-front peculiarities in strong anisotropic hyperbolic and σ-near-zero regimes of the metasurface. [ABSTRACT FROM AUTHOR]