The equations of state and structures of andalusite and sillimanite have been determined using high-pressure single-crystal X-ray diffraction. A third-order Birch-Murnaghan equation-of-state fit to 14 P-V data points measured between 1 bar and 9.8 GPa for andalusite yields values of KT0 = 144.2(7) GPa and K′ = 6.8(2). A similar analysis for sillimanite involving a fit to 13 P-V data points between 1 bar and 8.5 GPa results in KT0 = 164(1) GPa and K′ = 5.0(3). The axial compression of both structures is nonlinear and highly anisotropic (~60%) with the c-axis being the least compressible axis in both structures. The axial moduli determined with a parameterized form of the third-order Birch-Murnaghan equation of state are: Ka0 = 163(1) GPa, Kb0 = 113.1(7) GPa, and Kc0 = 297(1) GPa with K′a0 = 2.1(3), K′b0 = 5.08(19), and K′c0 = 11.1(4) for sillimanite, and Ka0 = 99.6(7) GPa, Kb0 = 152.2(9) GPa, and Kc0 = 236(3) GPa with K′a0 = 5.83(19), K′b0 = 7.6(3), and K′c0 = 5.5(9) for andalusite. The major compression mechanism in both structures involves shortening of bond lengths within the AlO6 octahedra with volume reductions of 7.4% and 5.1% in sillimanite and andalusite, respectively, over the pressure ranges studied. In andalusite there is also significant compression of the AlO5 polyhedra and, to a lesser degree, the SiO4 tetrahedra that display reductions of 5.0% and 3.1% in volume, respectively. In sillimanite there is no significant compression of either the AlO4 or SiO4 tetrahedra which behave as rigid, incompressible units.