The metal-insulator and spin state transitions of CoO under high pressure are studied by using density functional theory combined with dynamical mean-field theory. Our calculations predict that the metal-insulator transition in CoO is a typical orbital selective Mott transition, where the t2g orbitals of Co 3d shell become metallic firstly around 60 GPa while the eg orbitals still remain insulating until 170 GPa. Further studies of the spin states of Co 3d shell reveal that the orbital selective Mott phase in the intermediate pressure regime is mainly stabilized by the high-spin state of the Co 3d shell and the transition from this phase to the full metallic state is driven by the high-spin to low-spin transition of the Co 2+ ions. Our results are in good agreement with the most recent transport and x-ray emission experiments under high pressure.PACS numbers: 71.30.+h, 71.27.+a, 75.30.Wx, 91.60.Gf Although the Mott metal-insulator transition (MIT) has been studied extensively for decades, most of the works are focused on the single band Hubbard model, where the Mott transition is driven completely by the ratio of the local Coulomb interaction and band width. While most of the Mott MITs in realistic materials [1] involve more than one band where the transition is driven not only by the local Coulomb interaction but also by the distribution of the electrons among these bands. For example, the redistribution of the four electrons among three bands may lead to so-called orbital selective Mott transition (OSMT) [2][3][4][5][6][7] in the t 2g bands. On the other hand, in many systems the redistribution of the electrons among different bands is induced by the crossover in spin states, i.e. the high-spin (HS) to low-spin (LS) transition [8]. Therefore in realistic materials (e.g. in 3d transition metal compounds) the Mott MIT and spin state crossovers are closely related to each other [9][10][11][12][13][14][15].Recently the high pressure experiments on charge transfer insulator CoO revealed very interesting behaviors in both transport properties and x-ray emission spectroscopy (XES). The transport measurement [16] indicated that with the increment of pressure there are two transitions in resistivity, one happens around 60 GPa and the other one takes place around 130 GPa. While the room temperature XES measurements [17,18] on the similar sample show that the spin state of Co 2+ ions persist in the HS state all the way to 140 GPa, after which the crossover from HS to LS states happens. Hence the interplay between the HS-LS transition and the two-step Mott insulator transition becomes the key factor to understand the underlying physics in CoO.The pressure-driven MIT and magnetic moment collapse in transition metal oxides have been studied by first principles calculations widely [9-11, 13, 19, 20]. As to CoO, its magnetic state transition under pressure was first discussed [13] within the Stoner scenario by employing the local spin density approximation (LSDA) and generalized gradient approximation (GGA) approaches of d...