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.
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