We report the results of more than seven years of monitoring of PSR J0537-6910, the 16 ms pulsar in the Large Magellanic Cloud, using data acquired with the Rossi X-ray Timing Explorer. During this campaign the pulsar experienced 22 sudden increases in frequency ("glitches" - 21 with increases of at least eight microHz) amounting to a total gain of over six parts per million of rotation frequency superposed on its gradual spindown of nu-dot = -2 x 10(exp -l0) Hz /s. 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 per part per million (6.5 days per micro Hz), such that these intervals can be predicted to within a few days, an accuracy which has never before been seen in any other pulsar. There appears to be an upper limit of approximately 40 micro Hz for the size of glitches in all pulsars, with the 1999 April glitch of PSR J0537-6910 as the largest so far. The change of its spindown across the glitches, delta (nu-dot), appears to have the same hard lower limit of -1.5 x 10 (exp -13) Hz/s, as, again, that observed in all other pulsars. The spindown continues to increase in the long term, nu-dot = -10(exp -21) Hz / s(exp 2), and thus the timing age of PSR 505374910 (-0.5 nu nu-dot (exp -1) continues to decrease at a rate of nearly one year every year, consistent with movement of its magnetic moment away from its rotational axis by one radian every 10,000 years, or about one meter per year. PSR J0537-6910 was likely to have been born as a nearly-aligned rotator spinning at 75-80 Hz, with a absolute value of nu considerably smaller than its current value of 2x 10(exp -10) Hz per second. Its pulse profile consists of a single pulse which is found to be flat at its peak for at least 0.02 cycles. Glitch activity may grow exponentially with a timescale of 170 years nu nu-dot ((nu nu-dot)(sub crab))exp -l in all young pulsars.