The ultimate mechanical properties, as characterized here by the ideal strengths of Mg2SiO4 forsterite, have been calculated using first-principles calculations and generalized gradient approximation under tensile and shear loading. The ideal tensile strengths (ITS) and ideal shear strengths (ISS) are computed by applying homogeneous strain increments along high-symmetry directions ([100], [010], and [001]) and low index shear plane ((100), (010), and (001)) of the orthorhombic lattice. We show that the ultimate mechanical properties of forsterite are highly anisotropic, with ITS ranging from 12.1 GPa along [010] to 29.3 GPa along [100], and ISS ranging from 5.6 GPa for simple shear deformation along (100) to 11.5 GPa for shear along (010).