We report on more than 7 yr of monitoring of PSR J0537-6910, the 16 ms pulsar in the LMC, using data acquired with RXTE. During this campaign the pulsar experienced 23 sudden increases in frequency ("glitches") amounting to a total gain of over 6 ppm of rotation frequency superposed on its gradual spin-down of = -2 × 10-10 Hz s-1. The time interval from one glitch to the next obeys a strong linear correlation to the amplitude of the first glitch, with a mean slope of about 400 days ppm-1 (6.5 days μHz-1), such that these intervals can be predicted to within a few days, an accuracy that has never before been seen in any other pulsar. There appears to be an upper limit of ~40 μHz for the size of glitches in all pulsars, with the 1999 April glitch of PSR J0537-6910 the largest so far. The change of its spin-down across the glitches, Δ, appears to have the same hard lower limit, -1.5 × 10-13 Hz s-1, as that observed in all other pulsars. The spin-down continues to increase in the long term, = -10-21 Hz s-2, and thus the timing age of PSR J0537-6910 (-0.5ν-1) continues to decrease at a rate of nearly 1 yr every year, consistent with movement of its magnetic moment away from its rotational axis by 1 rad every 10,000 yr, or about 1 m yr-1. PSR J0537-6910 was likely to have been born as a nearly aligned rotator spinning at 75-80 Hz, with a || considerably smaller than its current value of 2 × 10-10 Hz s-1. Its pulse profile consists of a single pulse that is found to be flat at its peak for at least 0.02 cycles.