The Enskog kinetic equation for hard spheres is the only tractable theory with which the transport properties of a moderately dense gas can be studied. However, relatively little is known about its solutions outside the linear regime. In this paper two approximate nonlinear solutions of the Enskog equation for uniform shear flow are presented: a perturbative solution to second order in the shear rate and to fourth order in velocity moments and a ”nonperturbative” moment solution to all orders in the shear rate and to second order in the velocity moments. A comparison to the results of nonequilibrium molecular-dynamics simulations shows that the perturbative results give good estimates of the quadratic corrections to the pressure tensor while the nonperturbative solution gives a semiquantitative description of viscoelastic effects including shear thinning and the normal stresses over a wide range of shear rates. The relevance of these results to the construction of kinetic models of the Enskog equation is also discussed.