A high-pressure phase transition in iron free MgSiO3 perovskite, the most abundant Earth forming mineral phase, was discovered and reported in 2003-2004. Here, we summarize our theoretical and computational studies on this phase transition and on the physical properties of the newly found post-perovskite phase. The theoretical approach is based on density functional theory and on the plane-wave pseudopotential method. We focus here on the structural, elastic, vibrational, and thermodynamic properties of the post-perovskite phase. The predicted Clapeyron slope of this transition and several elastic properties of the highpressure phase strongly suggest that this new phase is an important candidate for a primary constituent in D″.