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We present a calculation of the profiles of emission lines originating in a relativistic, eccentric disk, and show examples of the resulting model profiles. Our calculations are motivated by the fact that in about one-quarter of the double-peaked emission lines observed in radio-loud active galactic nuclei (and in the mildly active nucleus of NGC 1097), the red peak is stronger than the blue peak, which is contrary to the prediction of relativistic, circular disk models. Using the eccentric disk model we fit some of the observed profiles that cannot be fitted with a circular disk model. We propose two possible scenarios for the formation of an eccentric disk in an active galactic nucleus: (a) tidal perturbation of the disk around a supermassive black hole by a smaller binary companion, and (b) formation of an elliptical disk from the debris resulting from the tidal disruption of a star by the central black hole. In the former case we show that the eccentricity can be long-lived because of the presence of the binary companion. In the latter case, although the inner parts of the disk may circularize quickly, we estimate that the outer parts will maintain their eccentricity for times much longer than the local viscous time. If any of the observed double-peaked emission lines do indeed arise in an eccentric disk, their profiles are likely to vary due to precession of the disk, thus providing a means of testing our proposed scenario. We estimate that for a black hole with a mass of order 10^6^ M_sun_, the precision period due to general relativistic advance of the pericenter can be as short as a decade. However, for a black hole with a mass of the order of 10^8^ M_sun_ the precession period is of the order of a few centuries, be it due to general relativistic effects or due to the tidal effects of a binary companion. We suggest that it may nevertheless be possible to detect profile variability on much shorter timescales by comparing the evolution of the line profile with detailed model predictions. We argue that line- profile variability may also be the most promising discriminant among competing models for the origin of asymmetric, double-peaked emission lines.