  We present a timing and glitch analysis of the young X-ray pulsar PSR
J0537$-$6910, located within the Large Magellanic Cloud, using 13 years of data
from the now decommissioned Rossi X-ray Timing Explorer. Rotating with a spin
period of 16 ms, PSR J0537$-$6910 is the fastest spinning and most energetic
young pulsar known. It also displays the highest glitch activity of any known
pulsar. We have found 42 glitches over the data span, corresponding to a glitch
rate of 3.2 yr$^{-1}$, with an overall glitch activity rate of $8.8\times
10^{-7}\,$yr$^{-1}$. The high glitch frequency has allowed us to study the
glitch behavior in ways that are inaccessible in other pulsars. We observe a
strong linear correlation between spin frequency glitch magnitude and wait time
to the following glitch. We also find that the post-glitch spin-down recovery
is well described by a single two-component model fit to all glitches for which
we have adequate input data. This consists of an exponential amplitude $A =
(7.6 \pm 1.0)\times 10^{-14}\,$s$^{-2}$ and decay timescale $\tau =
27_{-6}^{+7}\,$d, and linear slope $m = (4.1\pm 0.4)\times
10^{-16}\,$s$^{-2}\,$d$^{-1}$. The latter slope corresponds to a second
frequency derivative $\ddot{\nu} = (4.7\pm 0.5) \times 10^{-22}\,$s$^{-3}$,
from which we find an implied braking index $n=7.4 \pm 0.8$. We also present a
maximum-likelihood technique for searching for periods in event-time data,
which we used to both confirm previously published values and determine
rotation frequencies in later observations. We discuss the implied constraints
on glitch models from the observed behavior of this system, which we argue
cannot be fully explained in the context of existing theories.
