We have presented the formalism used to compute thermoelastic properties of solids using the statically constrained QHA (sc-QHA). Combined with first principles LDA calculations it is a simple and accurate although computationally intensive method to compute thermoelastic properties of crystals from which elasticity of aggregates can be obtained within bounds. Results for MgO, MgSiO3-perovskite, and MgSiO3-post-perovskite and comparisons with high (Figure Presented) Figure 14. Corrections to bulk and shear moduli of (a) pervskite (Wentzcovitch et al. 2004) and (b) postperovskite (Wentzcovitch et al. 2006). From Carrier et al. (2008). pressure and high temperature experimental data have been reviewed to illustrate the accuracy possible to achieve. An interesting byproduct of QHA calculations is the prediction of high temperature crystal structures. This is possible because, by construction, the QHA relates crystal structure parameters and (non-interacting) phonon frequencies uniquely with volume and such structure-volume relationship is well established by static calculations. In non-cubic solids, sc-QHA calculations develop deviatoric thermal stresses at high temperatures. Thermal pressure is not isotropic. Relaxation of these stresses leads to a series of corrections to the structure that may be taken to any desired order, up to self-consistency. This is the self-consistent QHA (sc-QHA). We have shown how to correct elastic constants for deviatoric stresses generated by the sc-QHA. We have illustrated the procedure by correcting to first order the elastic constants of perovskite and post-perovskite, the major silicate phases of the Earth's lower mantle. This correction is very satisfactory for obtaining the aggregate elastic constants and velocities of these minerals at in situ conditions of the lower mantle. This procedure can also be used to predict elastic constants in the presence of deviatoric stresses, or to correct elasticity measurements performed under non-hydrostatic conditions, as often happen in diamond-anvil cells, if deviatoric stresses are known. Several insights of geophysical significance have been obtained from high temperature elasticity calculations: 1) MgO is the most anisotropic phase in the lower mantle. Since it is also the weakest, it is a potentially important source of anisotropy in aggregates with lattice preferred orientation produced by mantle flow; 2) aggregates with typical pyrolite Mg/Si ratio, i.e., ∼20 vol% of MgO and ∼80 vol% of perovskite, appear to reproduce the elastic properties of the lower mantle better than the pure perovskite aggregate, at least down to ∼1,600 km depth. This suggests that shallow to mid lower mantle has the same chemistry of the upper mantle (as far as Mg/Si ratio is concerned); 3) in the deep lower mantle, post-perovskite free aggregates appear to slowly develop elastic properties that deviate from those of the Preliminary Reference Earth Model (PREM). The elastic properties of post-perovskite suggest that this deviation is caused by its presence in the D″ layer, the deepest ∼300 km of the mantle; 4) the post-perovskite transition causes velocity jumps similar to those detected in some places in D″ beneath subduction zones. This is consistent with expectations based on the post-perovskite phase boundary with positive Clapeyron slope; 5) velocity changes across the post-perovskite transition suggest that the anti- correlation between lateral bulk and shear velocity changes in D″ could be caused by lateral changes in the perovskite/post-perovskite abundances. For almost two decades the D″ region has remained an enigma. It is a complex layer at the interface of two chemically distinct regions, mantle and core. Post-perovskite should co-exist with other solid phases, and probably also with melts, in this region. There are still several mysteries, in the details, to be resolved; but, the insights offered by these calculations have advanced considerably knowledge of the puzzling D″ layer. These successes show that first principles calculations of thermoelastic properties of mantle minerals are poised to contribute much more to our understanding of the planet in the years ahead.