1 Introduction.- 2 Seismic-Wave Propagation in Anisotropic Media.- 2.1 A simple anisotropic medium: the stack of isotropic layers.- 2.2 Elastic tensors in general anisotropic media.- 2.3 Plane waves in homogeneous anisotropic media.- 2.3.1 Hexagonal symmetry case.- 2.3.2 Phase and group velocities.- 2.3.3 Perturbation theory.- 2.4 Body waves in heterogeneous anisotropic media.- 2.4.1 Ray theory.- 2.4.2 Computational problems.- 2.4.3 Concluding remarks.- 2.5 Surface waves in anisotropic media.- 2.5.1 Love and Rayleigh waves.- 2.5.2 Love/Rayleigh-wave incompatibility.- 2.5.3 Perturbation theory.- 2.5.4 Concluding remarks.- 3 Elastic Anisotropy of Rock-Forming Minerals and Polycrystalline RockS.- 3.1 Methods of investigation.- 3.2 Velocity anisotropy in rock-forming minerals.- 3.3 Elastic anisotropy of polycrystalline aggregates.- 3.3.1 Effects of cracks and grain boundaries.- 3.3.2 Effects of preferred mineral orientation.- 3.4 Crustal rocks.- 3.5 Uppermost mantle rocks.- 4 Interpretation of Seismic Data in Terms of Anisotropy.- 4.1 Trade-off between seismic anisotropy and heterogeneity.- 4.2 Seismic anisotropy and inverse methods.- 4.3 Interpretation of body-wave data.- 4.3.1 Refraction data.- 4.3.2 Shear-wave splitting.- 4.4 Long-range refraction profiles.- 4.5 Interpretation of surface-wave data.- 5 Anisotropic Structures in the Lithosphere.- 5.1 Oceanic crust.- 5.2 Subcrustal lithosphere under oceans.- 5.2.1 Observations.- 5.2.2 Causes and origin of anisotropy.- 5.2.3 Depth range of anisotropy in the oceanic lithosphere.- 5.3 Subduction zones.- 5.4 Continental crust.- 5.4.1 Upper crust.- 5.4.2 Lower crust.- 5.5 Subcrustal lithosphere of continents.- 5.5.1 Observations.- 5.5.2 Petrological constraints for interpretations.- 5.5.3 Anisotropie models.- 6 Anisotropy in Deeper Parts of the Earth.- 6.1 Asthenosphere.- 6.1.1 The depth extent of large-scale anisotropy.- 6.1.2 Transversely isotropic models.- 6.1.3 Azimuthal anisotropy.- 6.1.4 Possible causes of seismic anisotropy in the asthenosphere.- 6.2 Seismic anisotropy at greater depths.- 6.2.1 Upper-mantle transition region and lower mantle.- 6.2.2 Core-mantle boundary.- 6.2.3 Inner core.- 7 Conclusions and Outlook.- 7.1 Possible causes of the observed seismic anisotropy.- 7.2 Outlook.- References.