The effect of rotation on a homogeneous turbulent shear flow has been studied by means of a series of direct numerical simulations with different rotation numbers. The evolution of passive scalar fields with mean gradients in each of the three orthogonal directions in the flow was investigated in order to elucidate the effect of rotation on turbulent scalar transport. Conditions of the near-wall region of a boundary layer were approached by using a rapid shear and therefore, comparisons could be made with rapid distortion theory based on the linearized equations of the flow and scalar transport. Reynolds stresses, pressure–strain correlations and two-point velocity correlations were computed and turbulent structures were visualized. It is shown that rotation has a strong influence on the time development of the turbulent kinetic energy, the anisotropy of the flow and on the turbulent structures. Furthermore, rotation significantly affects turbulent scalar transport. The transport rate of the scalar and the direction of the scalar flux vector show large variations with different rotation numbers, and a strong alignment was observed between the scalar flux and the principal axes of the Reynolds stress tensor. The ratio of the turbulent and scalar time scales is influenced by rotation as well. The predictions of the linear theory of the turbulent one-point statistics and the scalar flux agreed fairly well with direct numerical simulation (DNS) results based on the full nonlinear governing equations. Nonetheless, some clear and strong nonlinear effects are observed in a couple of cases which significantly influence the development of the turbulence and scalar transport.