Forsterite, Mg2SiO4, the magnesian end-member of the olivine system, is the archetypal example of an orthosilicate structure. We have conducted synchrotron-based single-crystal X-ray diffraction experiments to 90 GPa on synthetic end-member forsterite to study its equation of state and phase transitions. Upon room-temperature compression, the forsterite structure is observed to 48 GPa. By fitting a third-order Birch-Murnaghan equation of state to our compression data, we obtain the zero-pressure isothermal bulk modulus, K0T = 130.0(9) GPa and its pressure derivative, K′0T = 4.12(7) for a fixed room-pressure volume, V0 = 290.1(1) Å3, in good agreement with earlier work. At 50 GPa, a phase transition to a new structure (forsterite II) occurs, followed by a second transition to forsterite III at 58 GPa. Forsterite III undergoes no additional phase transitions until at least 90 GPa. There is an ~4.8% volume reduction between forsterite and forsterite II, and a further ~4.2% volume reduction between forsterite II and III. On decompression forsterite III remains until as low as 12 GPa, but becomes amorphous at ambient conditions. Using our X-ray diffraction data together with an evolutionary crystal structure prediction algorithm and metadynamics simulations, we find that forsterite II has triclinic space group P1 and forsterite III has orthorhombic space group Cmc21. Both high-pressure phases are metastable. Metadynamics simulations show a stepwise phase transition sequence from 4-coordinated Si in forsterite to mixed tetrahedral and octahedral Si (as in forsterite II), and then fully sixfold-coordinated Si (as in forsterite III), occurring by displacement in (100). The forsterite III structure is a member of the family of post-spinel structures adopted by compositions such as CaFe2O4 and CaTi2O4.