We present reverberation mapping (RM) results for 17 high-redshift, high-luminosity quasars with good-quality R-band and emission-line light curves. We are able to measure statistically significant lags for Ly alpha (11 objects), Si IV (5 objects), C IV (11 objects), and C III] (2 objects). Using our results and previous lag determinations taken from the literature, we present an updated C IV radius-luminosity relation and provide for the first time radius- luminosity relations for Ly alpha, Si IV, and C III]. While in all cases the slopes of the correlations are statistically significant, the zero points are poorly constrained because of the lack of data at the low-luminosity end. We find that the emissivity-weighted distances from the central source of the Ly alpha, Si IV, and C III] line-emitting regions are all similar, which corresponds to about half that of the H beta region. We also find that 3/17 of our sources show an unexpected behavior in some emission lines, two in the Ly alpha light curve and one in the Si IV light curve, in that they do not seem to follow the variability of the UV continuum. Finally, we compute RM black hole (BH) masses for those quasars with highly significant lag measurements and compare them with C IV single-epoch (SE) mass determinations. We find that the RM-based BH mass determinations seem smaller than those found using SE calibrations.