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Reverberation mapping of nearby active galactic nuclei has led to estimates of broad-line-region (BLR) sizes and central-object masses for some 37 objects to date. However, successful reverberation mapping has yet to be performed for quasars of either high luminosity (above L_opt~10^{46} erg/s) or high redshift (z>0.3). Over the past six years, we have carried out, at the Hobby-Eberly Telescope, rest-frame-ultraviolet spectrophotometric monitoring of a sample of six quasars at redshifts z=2.2--3.2, with luminosities of L_opt~10^{46.4}--10^{47.6} erg/s, an order of magnitude greater than those of previously mapped quasars. The six quasars, together with an additional five having similar redshift and luminosity properties, were monitored photometrically at the Wise Observatory during the past decade. All 11 quasars monitored show significant continuum variations of order 10%--70%. This is about a factor of two smaller variability than for lower luminosity quasars monitored over the same rest-frame period. In the six objects which have been spectrophotometrically monitored, significant variability is detected in the CIV1550 broad emission line. In several cases the variations track the continuum variations in the same quasar, with amplitudes comparable to, or even greater than, those of the corresponding continua. In contrast, no significant Ly\alpha variability is detected in any of the four objects in which it was observed. Thus, UV lines may have different variability trends in high-luminosity and low-luminosity AGNs. For one quasar, S5~0836+71 at z=2.172, we measure a tentative delay of 595 days between CIV and UV-continuum variations, corresponding to a rest-frame delay of 188 days and a central black-hole mass of 2.6\times10^9 M_\odot.