<p>A fourth-order classical Stokes theory for periodic gravity waves propagating steadily on linear shear currents in a fluid of finite depth is presented. The primary objective of this paper is the computation of the well-known Stokes wave expansion to fourth-order and to assess the validity of the fourth-order solution by comparison with the exact results. The main findings of this paper are (a) the wave velocity is strongly dependent on linear shear currents, (b) to find the effects of depth uniform current on the wave profiles using fourth order results, and (c) different wave profiles are found and the influence of shear currents on these profiles is shown to be significant. For the steepest waves, it is found that the new fourth-order analysis gives considerable differences on the profiles when compared with the third-order analysis, and that the fourth-order results are more compatible with the numerical results. The results in the deep-water limit are also deduced and discussed.</p>