In this article, we consider a problem of the surface elastic wave propagation within the framework of the isotropic Cosserat continuum. The medium deformation in this model is described not only by the displacement vector but also by a kinematically independent rotation vector. We discuss the general solution of equations of motion. This solution describes the following wave types: longitudinal and transverse bulk waves, Rayleigh wave, surface transverse wave in a half-space as well as Lamb wave and transverse wave in a thin layer. Within the framework of Cosserat continuum, both the Rayleigh and surface transverse waves in a half-space are dispersive. The transverse wave in a thin layer and the surface transverse wave in a half-space do not have any analogies in the classical elasticity theory.