In my work, magnetic and structural transitions of three categories of compounds are investigated by density functional calculations under local spin density approximation (LSDA). The first compound is Rh2MnGe with full Heusler structure at ambient condition. However, the structure is unstable at T=0 according to our calculations. A more stable structure we found is tetragonal one with either extension or compression along c-axis. The electronic reason of this distortion is the band Jahn-Teller effect where the Jahn-Teller active states are 4d states of Rh which is accidently put at the Fermi level by spin splitting. Then, magnetic moment behavior under pressure in itinerant compounds is investigated in four cubic Laves phase compounds (YFe2, ZrFe2, HfFe2, and LuFe2). The magnetic spin moment is decreased under pressure. A magnetic collapse where the spin moment vanishes is predicted under pressure around 20 GPa for Zr and Hf compounds, 40 GPa for Y and Lu compounds. The behavior of the magnetic moment is the result of competition between magnetic exchange interactions and kinetic energy during the compression of the volume, as described by the Stoner model. The last material investigated is CoO using LDSA+U in order to describe the strong Coulombic interaction of the transition metal ion. The pressure induced a magnetic transition, which was discovered in experiments, is explained by the competition between ligand field splitting and exchange energy. The ligand field splitting is increased under pressure, and suppresses the intraatomic exchange. As a result, the spin state changes from high spin to low spin, and at last to nonmagnetic state.