The generalized Lagrangian mean (GLM) formulation is used to describe the interaction of waves and currents. In contrast to the more conventional Eulerian formulation the GLM description enables splitting of the mean and oscillating motion over the whole depth in an unambiguous and unique way, also in the region between wave crest and trough. The present paper deals with non-breaking long-crested regular waves on a current using the GLM formulation coupled with a WKBJ-type perturbation-series approach. The waves propagate under an arbitrary angle with the current direction. The primary interest concerns nonlinear changes in the vertical distribution of the mean velocity due to the presence of the waves, but modifications of the orbital velocity profiles, due to the presence of a current, are considered as well. The special case of no initial current, where waves induce a so-called drift velocity or mass-transport velocity, is also studied.