A theoretical study of vibrational dephasing of molecular vibrations in Liquids is presented with an aim to understand the experimentally observed sub-quadratic quantum number (n) dependence bf the vibrational dephasing rate, in systems like CH3I and CHCl3 and their deuterated analogues. The analysis is based on Oxtoby's theory of vibrational dephasing but with a detailed microscopic description of the frequency dependent frictional forces on the vibrational mode. The friction on the normal coordinate in liquids is found to have a pronounced biphasic behavior with a dominant Gaussian initial component followed by a slow exponential-like relaxation. While the exponential relaxation usually assumed in Kubo's stochastic theory leads to a quadratic n dependence of the dephasing rate, the biphasic friction is shown to give rise to the sub-quadratic n dependence. As the biphasic frictional response is expected to be a generic feature of the friction on any vibrational coordinate in dense liquids, the sub-quadratic quantum number dependence is predicted to be common to most ultrafast overtone dephasing. In addition, the calculated rates (without any adjustable parameter), are found to be in good agreement with the experimental results for the C-I stretching mode in liquid CH3I and for the C-H stretching in liquid CHCl3. (C) 1997 American Institute of Physics.