First principle calculations were performed to investigate the structural, mechanical, electronic properties, and thermodynamic properties of three binary Mg-B compounds under pressure, by using the first principle method. The results implied that the structural parameters and the mechanical properties of the Mg-B compounds without pressure are well matched with the obtainable theoretically simulated values and experimental data. The obtained pressure-volume and energy-volume revealed that the three Mg-B compounds were mechanically stable, and the volume variation decreases with an increase in the boron content. The shear and volume deformation resistance indicated that the elastic constant Cij and bulk modulus B increased when the pressure increased up to 40 GPa, and that MgB7 had the strongest capacity to resist shear and volume deformation at zero pressure, which indicated the highest hardness. Meanwhile, MgB4 exhibited a ductility transformation behaviour at 30 GPa, and MgB2 and MgB7 displayed a brittle nature under all the considered pressure conditions. The anisotropy of the three Mg-B compounds under pressure were arranged as follows: MgB4 > MgB2 > MgB7. Moreover, the total density of states varied slightly and decreased with an increase in the pressure. The Debye temperature ΘD of the Mg-B compounds gradually increased with an increase in the pressure and the boron content. The temperature and pressure dependence of the heat capacity and the thermal expansion coefficient α were both obtained on the basis of Debye model under increased pressure from 0 to 40 GPa and increased temperatures. This paper brings a convenient understanding of the magnesium-boron alloys.