We present the results from a study of the dynamics of the system of globular clusters around M87. After eliminating foreground galactic stars and background galaxies, we end up with a sample of 205 bona fide M87 globular clusters for which we have radial velocities determined from multislit spectra taken with the Low Resolution Imaging Spectrograph on the Keck Telescope. We find that the mean radial velocity of the M87 globular clusters agrees well with that of M87 itself and that the velocity histogram is well represented by a Gaussian distribution. We find evidence for rotation in the globular cluster system. We find that the observed velocity dispersion of the M87 globular cluster system increases with radius from 270 km s^-1 atr = 9 kpc to ≈ 400 km s^-1 atr = 40 kpc. The inferred mass-to-light ratio in solar units increases from 5 atr = 9 kpc to ≈ 30 atr = 40 kpc withM(r) ~r^1.7. The long-slit optical spectroscopy near the center of M87 and the recent analysis of theROSAT X-ray data are in good agreement with this analysis near the nucleus and in the outer parts of M87, respectively.