Using a nonlocal nematic potential, we generalize the Doi theory for nematic polymers to include distortional elasticity. We derive an evolution equation for the configuration tensor and a constitutive equation for a nonlocal stress tensor which is consistent with the long-range order in nematic polymers. One of the interesting effects of distortional elasticity is the appearance of a mean-field torque on the molecules as they are forced away by flow from their preferred orientation. This torque gives rise to an antisymmetric part of the stress tensor. With a few molecular parameters, the complete system of equations is capable, we believe, of describing the evolution of the texture and the dynamics of disclinations in flowing nematic polymers. Thus, for the first time, a suitable platform for exploring complex flows of nematic polymers is established. In the limit of weak flows and small distortions, the theory properly reduces to the Leslie-Ericksen theory. The Leslie viscosities are derived in terms of molecular parameters.