By means of a combination of equilibrium Monte Carlo and molecular dynamics simulations and nonequilibrium molecular dynamics we investigate the ordered, uniaxial phases (i.e., nematic and smectic A) of a model liquid crystal. We characterize equilibrium behavior through their diffusive behavior and elastic properties. As one approaches the equilibrium isotropic-nematic phase transition, diffusion becomes anisotropic in that self-diffusion D⊥ in the direction orthogonal to a molecule's long axis is more hindered than self-diffusion D∥ in the direction parallel to that axis. Close to nematic-smectic A phase transition the opposite is true, D∥ < D⊥. The Frank elastic constants K1, K2, and K3 for the respective splay, twist, and bend deformations of the director field n̂ are no longer equal and exhibit a temperature dependence observed experimentally for cyanobiphenyls. Under nonequilibrium conditions, a pressure gradient applied to the smectic A phase generates Poiseuille-like or plug flow depending on whether the convective velocity is parallel or orthogonal to the plane of smectic layers. We find that in Poiseuille-like flow the viscosity of the smectic A phase is higher than in plug flow. This can be rationalized via the velocity-field component in the direction of the flow. In a sufficiently strong flow these smectic layers are not destroyed but significantly bent.