We review millimeter interferometric phase variations caused by variations in the precipitable water vapor content of the troposphere, and we discuss techniques proposed to correct for these variations. We present observations with the Very Large Array (VLA) at 22 and 43 GHz designed to test these techniques. We find that both the fast switching and paired array calibration techniques are effective at reducing tropospheric phase noise for radio interferometers. In both cases, the residual rms phase fluctuations after correction are independent of baseline length for b>beff. These techniques allow for diffraction-limited imaging of faint sources on arbitrarily long baselines at millimeter wavelengths. We consider the technique of tropospheric phase correction using a measurement of the precipitable water vapor content of the troposphere via a radiometric measurement of the brightness temperature of the atmosphere. Required sensitivities range from 20 mK at 90 GHz to 1 K at 185 GHz for the millimeter array (MMA) and to 120 mK for the VLA at 22 GHz. The minimum gain stability requirement is 200 at 185 GHz at the MMA, assuming that the astronomical receivers are used for radiometry. This increases to 2000 for an uncooled system. The stability requirement is 450 for the cooled system at the VLA at 22 GHz. To perform absolute radiometric phase corrections also requires knowledge of the tropospheric parameters and models to an accuracy of a few percent. It may be possible to perform an empirically calibrated radiometric phase correction, in which the relationship between fluctuations in brightness temperature differences and fluctuations in interferometric phases is calibrated by observing a strong celestial calibrator at regular intervals. A number of questions remain concerning this technique, including the following: (1) Over what timescale and distance will this technique allow for radiometric phase corrections when switching between the source and the calibrator? (2) How often will calibration of the TBrms-ϕrms relationship be required? ¿ 1999 American Geophysical Union