Hydrodynamic equations of motion for a monodisperse collection of nearly smooth homogeneous spheres have been derived from the corresponding Boltzmann equation, using a Chapman-Enskog expansion around the elastic smooth spheres limit. Because in the smooth limit the rotational degrees of freedom are uncoupled from the translational ones, it turns out that the required hydrodynamic fields include (in addition to the standard density, velocity, and translational granular temperature fields) the (infinite) set of number densities, n(s,r, t), corresponding to the continuum of values of the angular velocities. The Chapman-Enskog expansion was carried out to high (up to 10th) order in a Sonine polynomial expansion by using a novel computer-aided method. One of the consequences of these equations is that the asymptotic spin distribution in the homogeneous cooling state for nearly smooth, nearly elastic spheres, is highly non-Maxwellian. The simple sheared flow possesses a highly non-Maxwellian distribution as well. In the case of wall-bounded shear, it is shown that the angular velocity injected at the boundaries has a finite penetration length.