Seismic anisotropy in the mantle is primarily due to deformation-induced lattice preferred orientation (LPO) of olivine crystals. The aim of this paper is to better understand how such LPO is produced by flow, and to learn whether observations of seismic anisotropy can be used to infer the local direction of flow within the mantle. The basis of the work is a two-dimensional theory which describes how LPO evolves during an arbitrary time-dependent deformation [Ribe, 1989]. I investigate the evolution of LPO beneath ocean ridges, above subducted slabs, and in thermal convective flows. The ocean ridge model predicts that the seismic anisotropy produced at a fast spreading ridge should be greater in magnitude than that produced at a slow spreading ridge, and that the fast axis should be more nearly horizontal. The subduction zone model predicts that strong seismic anisotropy should be produced above subducted slabs, with the fast axis aligned with the slab, regardless of the subduction angle. The evolution of LPO in typical convective flows is extremely complex, and depends strongly on such factors as the mode of heating and the surface boundary conditions. In general, there is no simple relation between the LPO and the local flow direction in any of the models considered. The attempt to use seismic anisotropy to map flow patterns on a global scale is therefore probably unwarranted. However, the LPO does become aligned with the flow direction when the flow is a progressive simple shear. Such a flow occurs in limited portions of the mantle: beneath the lithosphere at a fast spreading ridge, above subducted slabs, and in boundary layers beneath rigid surfaces such as continents. Observations of seismic anisotropy may therefore indicate the sense of shear at shallow depths in the mantle.