We present the lightcurves of 21 gravitational microlensing events from the first six years of the MACHO Project gravitational microlensing survey which are likely examples of lensing by binary systems. These events were manually selected from a total sample of ~350 candidate microlensing events which were either detected by the MACHO Alert System or discovered through retrospective analyses of the MACHO database. At least 14 of these 21 events exhibit strong (caustic) features, and 4 of the events are well fit with lensing by large mass ratio (brown dwarf or planetary) systems, although these fits are not necessarily unique. The total binary event rate is roughly consistent with predictions based upon our knowledge of the properties of binary stars, but a precise comparison cannot be made without a determination of our binary lens event detection efficiency. Towards the Galactic bulge, we find a ratio of caustic crossing to non-caustic crossing binary lensing events of 12:4, excluding one event for which we present 2 fits. This suggests significant incompleteness in our ability to detect and characterize non-caustic crossing binary lensing. The distribution of mass ratios, N(q), for these binary lenses appears relatively flat. We are also able to reliably measure source-face crossing times in 4 of the bulge caustic crossing events, and recover from them a distribution of lens proper motions, masses, and distances consistent with a population of Galactic bulge lenses at a distance of 7 +/- 1 kpc. This analysis yields 2 systems with companions of ~0.05 M_sun.
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