© 2017 Elsevier B.V. Seismic anisotropy has been found in many regions of the Earth's interior. Its presence in the Earth's crust has been known since the 19th century, and is due in part to the alignment of anisotropic crystals in rocks, and in part to patterns in the distribution of fractures and pores. In the upper mantle, seismic anisotropy was discovered 50 years ago, and can be attributed for the most part, to the alignment of intrinsically anisotropic olivine crystals during large scale deformation associated with convection. There is some indication for anisotropy in the transition zone, particularly in the vicinity of subducted slabs. Here we focus on the deep Earth – the lower mantle and core, where anisotropy is not yet mapped in detail, nor is there consensus on its origin. Most of the lower mantle appears largely isotropic, except in the last 200–300 km, in the D″ region, where evidence for seismic anisotropy has been accumulating since the late 1980s, mostly from shear wave splitting measurements. Recently, a picture has been emerging, where strong anisotropy is associated with high shear velocities at the edges of the large low shear velocity provinces (LLSVPs) in the central Pacific and under Africa. These observations are consistent with being due to the presence of highly anisotropic MgSiO3 post-perovskite crystals, aligned during the deformation of slabs impinging on the core-mantle boundary, and upwelling flow within the LLSVPs. We also discuss mineral physics aspects such as ultrahigh pressure deformation experiments, first principles calculations to obtain information about elastic properties, and derivation of dislocation activity based on bonding characteristics. Polycrystal plasticity simulations can predict anisotropy but models are still highly idealized and neglect the complex microstructure of polyphase aggregates with strong and weak components. A promising direction for future progress in understanding the origin of seismic anisotropy in the deep mantle and its relation to global mantle circulation, is to link macroscopic information from seismology and microscopic information mineral physics through geodynamics modeling. Anisotropy in the inner core was proposed 30 years ago to explain faster P wave propagation along the direction of the Earth's axis of rotation as well as anomalous splitting of core sensitive free oscillations. There is still uncertainty about the origin of this anisotropy. In particular, it is difficult to explain its strength, based on known elastic properties of iron, as it would require almost perfect alignment of iron crystals. Indeed, the strongly anomalous P travel times observed on paths from the South Sandwich Islands to Alaska may or may not be due to inner core anisotropy, and will need to be explained before consensus can be reached on the strength of anisotropy in the inner core and its origin.