Rocks often develop fabric when subject to deformation, and this fabric causes anisotropy of physical properties, including effective viscosity and seismic velocities. While predicting anisotropy from mantle flow models is becoming increasingly popular, the effect of viscous anisotropy on the flow has yet to be investigated thoroughly. We investigated two classes of mantle flow - subduction and Rayleigh-Taylor instability, using analytical and numerical solutions. For both flow classes, we find that including anisotropic viscosity strongly modifies the flow. Our results have far-reaching implications, for example on the thermal structure and melting regime in subduction zones or the length scale and timing of lithospheric delamination.