Quasiharmonic theory combined with first-principles phonon density of states gives accurate thermodynamics properties of minerals at the high pressures and temperatures of the Earth interior. Care must be exercised in using this method within its regime of validity. A simple criterion to establish the approximate upper temperature limit of validity of the QHA versus pressure was introduced based on a posteriori inspection of the thermal expansivity. This criterion shows that the QHA is a good approximation for minerals at mantle conditions, except for truly anharmonic crystals like CaSiO3- perovskite, and perhaps for minerals at conditions of the core-mantle boundary. In general, the temperature range of applicability of the QHA increases with pressure. The systems analyzed here, were all investigated consistently and systematically, using the same pseudopotentials, convergence criteria, plane-wave cut-offs, and k- and q-point samplings in LDA and PBE-GGA calculations. The trends extracted from these calculations are therefore reliable. The importance of zero-point-motion effects on structural properties cannot be overemphasized. The LDA exchange-correlation functional gives considerably superior results for 0 K properties after the zero-point-motion energy is included in the calculation. GGA results consistently overestimate volume and underestimate the bulk modulus by several percent. LDA thermodynamic properties at ambient conditions are in excellent agreement with experimental values. The performance of these exchange-correlation functionals was also investigated for predictions of thermodynamic phase boundaries. In general, the LDA phase boundaries are "shifted" by 5-10 GPa to lower pressures compared to the experimental ones, while GGA boundary are closer to experimental ones but still "shifted" by 2-5 GPa to higher pressures. LDA and GGA Clapeyron slopes are very similar and in fairly good agreement with experimental slopes. Phase boundaries, however, may be more affected by anharmonicity since bond-breaking and bond-reconstruction, phonon softening, diffusion, etc may take place at these pressures and temperatures. Discontinuities in density and bulk sound velocity for important phase transformations in the mantle transition zone were systematically investigated. Only the magnesium endmembers, Mg 2SiO4, were studied. Predicted discontinuities in density, bulk modulus, and bulk sound velocity are sharp and have useful accuracy for analysis of mantle discontinuities, despite uncertainties in the predicted phase boundary. With the rough premise that the olivine system does not interact with other minerals such as pyroxene/garnet/Ca-perovskite, we were able estimate density discontinuities at 410-km, 520-km, and 660-km depth and compare them with those inferred from seismic data. We conclude that seismic density discontinuities observed at 410-km (0.2-4%) and 660-km (∼5.2%) depth can be produced by phase transition in the olivine system alone in a mantle with pyrolite composition (~60 vol% olivine). However, the 520-km discontinuity (2.1±0.8%) cannot. It requires contributions from other mantle transformations, e.g., the Ca-perovskite ex-solution from majorite garnet. We also discuss the post-perovskite phase boundary. Our predicted Clapeyron slope, ∼7.5±0.3 MPa/K, differs somewhat from the preferred experimental slopes, > 9.7 MPa/K. This suggests that this phase boundary might be sensitive to anharmonic effects. We have introduced a semi-empirical ansatz to compute anharmonic contributions to the free energy. This method utilizes experimental data at low pressures and high temperatures on one property, preferably thermal expansivity, to compute all thermodynamic properties at high pressures and high temperatures. It offered excellent results for MgO, and the P-V-T relation in this mineral was offered for pressure calibration in diamond-anvil-cells experiments. It was also applied to forsterite (α), wadsleyite (β), and to the α-to-β transformation. The thermodynamic properties of the α- and β-phases improve and are further reconciled with experimental measurements beyond the QHA validity regime after correction for anharmonic effects. This study indicated that anharmonicity manifests differently in different systems, depending whether the "average" phonon frequency increases (β and MgO) or decreases (α) with temperature at constant volume. This difference in behavior affects the Clapeyron slope of the α-to-β transformation, raising it from 2.5-2.6 to 3.6 MPa/K and reconciling it with the latest experimental determinations. Anharmonic effects are most evident in the thermal expansivity, followed by the thermal Grüneisen parameter, constant pressure specific heat, and least evident in the bulk modulus.