Violation of Lorentz invariance in nature is a possibility suggested by various candidate theories of quantum gravity and exotic extensions of the standard model, and by general curiosity. Lorentz-violating effects in particle interactions are strongly constrained, but effects involving gravity are not. Here, observational constraints on, and theoretical aspects of, a certain class of gravitational theories that violate Lorentz invariance are considered. "Einstein-aether" theory is a four-parameter class of theories in which gravity couples to a dynamical, timelike, unit-norm vector field: the "aether". This family provides a means for studying Lorentz violation in a generally covariant setting. Demonstrated first is the effect on the four parameters, of stretching the metric along the aether direction. Next, the Noether charge method for defining the Hamiltonian of a diffeomorphism invariant field theory is applied to obtain expressions for the total energy, momentum, and angular momentum of an Einstein-aether spacetime. The method is also used to discuss the mechanics of Einstein-aether black holes. Next, the computation of the theory's post-Newtonian parameters are reported. Constraints on their values are combined with other constraints concerning the properties of linearized wave modes and Einstein-aether cosmology. All of these constraints are satisfied by parameters in a large two-dimensional region in the theory's four-dimensional parameter space. Next, constraints from the motion of binary pulsar systems are considered. Derived to lowest post-Newtonian order are wave forms for the metric and aether far from a nearly Newtonian system and the rate of energy radiated by the system, in the limit that effects due to strong fields are neglected. There exists a one-parameter family of Einstein-aether theories for which the radiation rate expression is identical to that of general relativity to the order worked to here. Finally, strong field effects are included by treating the compact bodies as point particles with nonstandard, velocity dependent interactions. Precise constraints cannot be stated for general parameter values until the values of the coupling coefficients of the nonstandard interactions can be calculated for a given stellar source. It is argued, though, that if the parameters are smaller than roughly (0.1), then all current observational tests impose just the three conditions that guarantee agreement in the weak field limit. Thus, there exists a family of Einstein-aether theories, with one mildly bounded free parameter, that satisfy the collected constraints.