The nonequilibrium dynamics of a binary Lennard-Jones mixture in a simple shear flow is investigated by means of molecular dynamics simulations. The range of temperature T investigated covers both the liquid, supercooled, and glassy states, while the shear rate gamma covers both the linear and nonlinear regimes of rheology. The results can be interpreted in the context of a nonequilibrium, schematic mode-coupling theory developed recently, which makes the theory applicable to a wide range of soft glassy materials. The behavior of the viscosity eta(T,gamma) is first investigated. In the nonlinear regime, strong shear-thinning is obtained, etasimilar togamma(-alpha(T)), with alpha(T)similar or equal to 2/3 in the supercooled regime. Scaling properties of the intermediate scattering functions are studied. Standard "mode-coupling properties" of factorization and time superposition hold in this nonequilibrium situation. The fluctuation-dissipation relation is violated in the shear flow in a way very similar to that predicted theoretically, allowing for the definition of an effective temperature T-eff for the slow modes of the fluid. Temperature and shear rate dependencies of T-eff are studied using density fluctuations as an observable. The observable dependence of T-eff is also investigated. Many different observables are found to lead to the same value of T-eff, suggesting several experimental procedures to access T-eff. It is proposed that a tracer particle of large mass m(tr) may play the role of an "effective thermometer." When the Einstein frequency of the tracers becomes smaller than the inverse relaxation time of the fluid, a nonequilibrium equipartition theorem holds with =k(B)T(eff), where v(z) is the velocity in the direction transverse to the flow. This last result gives strong support to the thermodynamic interpretation of T-eff and makes it experimentally accessible in a very direct way.