This thesis makes contributions to the methodology of quantitative description of the tilting systems of perovskite structures and theoretical analysis of high-pressure phase transitions of representative perovskites. Chapter 1 and 2 introduce the perovskite structures, tilting classification and descriptions. The structures in each of the 15 tilt systems have been decomposed in to the amplitudes of symmetry-adapted modes in order to provide a clear and unambiguous definition of the tilt angles. A general expression in terms of tilt angles for the ratio of the volumes of the two polyhedra within the perovskite structure is derived. Chapter 3 uses the first-principles plane-wave pseudopotential calculations to investigate the high-pressure to phase transition and elasticity of LaAlO3 perovskite. This second order transition is determined to occur at ~14 GPa by the pressure variation of the squared frequency of the soft R-point mode in the structure. Elastic moduli are inverted from the calculated stress-strain data by the singular value decomposition method. The Landau parameters for this phase transition are calibrated from the calculation results. Chapter 4 uses the same method to investigate the high-pressure phase transitions and elasticity of YAlO3 perovskite. The Pnma, Imma, I4/mcm, , perovskite structures and the NH4CdCl3-, Gd2S3-, U2S3-, CaIrO3-type structures are considered. A continuous Pnma to Imma transition occurs at ~89 GPa, determined from the soft Z-point mode of the Imma structure. Then, a discontinuous Imma to I4/mcm transition occurs at ~100 GPa, suggested by the relative enthalpies and phonon dispersions. The elasticities of the Pnma, Imma and I4/mcm structures show mechanical stabilities compatible with the phase transitions. The NH4CdCl3- and CaIrO3-type structures are dynamically stable although not energetically favorable. The relative A/B site polyhedral volume ratios are found to qualitatively reproduce the relative enthalpies of the perovskite structures. / Ph. D.