We present results from 13776 radial-velocity (RV) measurements of 1278 candidate members of the old (4Gyr) open cluster M67 (NGC 2682). The measurements are the results of a long-term survey that includes data from seven telescopes with observations for some stars spanning over 40years. For narrow-lined stars, RVs are measured with precisions ranging from about 0.1 to 0.8km/s. The combined stellar sample reaches from the brightest giants in the cluster down to about 4mag below the main-sequence turnoff (V=16.5), covering a mass range of about 1.34M_{sun}_ to 0.76M_{sun}_. Spatially, the sample extends to a radius of 30arcmin (7.4pc in projection at a distance of 850pc or 6-7 core radii). We find M67 to have a mean radial velocity of +33.64km/s (with an internal precision of +/-0.03km/s) well separated from the mean velocity of the field. For stars with >=3 measurements, we derive RV membership probabilities and identify RV variables, finding 562 cluster members, 142 of which show significant RV variability. We use these cluster members to construct a color-magnitude diagram and identify a rich sample of stars that lie far from the standard single star isochrone, including the well-known blue stragglers, sub-subgiants and yellow giants. These exotic stars have a binary frequency of (at least) 80%, more than three times that detected for stars in the remainder of the sample. We confirm that the cluster is mass segregated, finding the binaries to be more centrally concentrated than the single stars in our sample at the 99.8% confidence level (and at the 98.7% confidence level when only considering main-sequence stars). The blue stragglers are centrally concentrated as compared to the solar-type main-sequence single stars in the cluster at the 99.7% confidence level. Accounting for measurement precision, we derive an RV dispersion in M67 of 0.80+/-0.04km/s for our sample of single main-sequence stars, subgiants and giants with V{<=}15.5. When corrected for undetected binaries, this sample yields a true RV dispersion of 0.59_-0.06_^+0.07^km/s. The radial distribution of the velocity dispersion is consistent with an isothermal distribution within our stellar sample. Using the cluster RV dispersion, we estimate a virial mass for the cluster of 2100_-550_^+610^M_{sun}_.