  In the course of conducting a deep (14.5 ~< r ~< 23), 20 night survey for
transiting planets in the rich ~550 Myr old open cluster M37 we have measured
the rotation periods of 575 stars which lie near the cluster main sequence,
with masses 0.2 Msun ~< M ~< 1.3 Msun. This is the largest sample of rotation
periods for a cluster older than 500 Myr. Using this rich sample we investigate
a number of relations between rotation period, color and the amplitude of
photometric variability. Stars with M >~ 0.8 Msun show a tight correlation
between period and mass with heavier stars rotating more rapidly. There is a
group of 4 stars with P > 15 days that fall well above this relation, which, if
real, would present a significant challenge to theories of stellar angular
momentum evolution. Below 0.8 Msun the stars continue to follow the period-mass
correlation but with a broad tail of rapid rotators that expands to shorter
periods with decreasing mass. We combine these results with observations of
other open clusters to test the standard theory of lower-main sequence stellar
angular momentum evolution. We find that the model reproduces the observations
for solar mass stars, but discrepancies are apparent for stars with 0.6 ~< M ~<
1.0 Msun. We also find that for late-K through early-M dwarf stars in this
cluster rapid rotators tend to be bluer than slow rotators in B-V but redder
than slow rotators in V-I_{C}. This result supports the hypothesis that the
significant discrepancy between the observed and predicted temperatures and
radii of low-mass main sequence stars is due to stellar activity.
