Inelastic scattering computations are presented for collisions of vibrationally and rotationally excited CO with H2 in full dimension. The computations utilize a newly developed six-dimensional potential energy surface (PES) and the previously reported four-dimensional V12 PES [P. Jankowski et al., J. Chem. Phys. 138, 084307 (2013)] and incorporate full angular-momentum coupling. At low collision energies, pure rotational excitation cross sections of CO by para-, ortho-, and normal-H2 are calculated and convolved to compare with recent measurements. Good agreement with the measured data is shown except for j1 = 0 → 1 excitation of CO for very low-energy para-H2 collisions. Rovibrational quenching results are presented for initially excited CO(v1j1) levels with v1 = 1, j1 = 1-5 and v1 = 2, j1 = 0 for collisions with para-H2 (v2 = 0, j2 = 0) and ortho-H2 (v2 = 0, j2 = 1) over the kinetic energy range 0.1-1000 cm(-1). The total quenching cross sections are found to have similar magnitudes, but increase (decrease) with j1 for collision energies above ∼300 cm(-1) (below ∼10 cm(-1)). Only minor differences are found between para- and ortho-H2 colliders for rovibrational and pure rotational transitions, except at very low collision energies. Likewise, pure rotational deexcitation of CO yields similar cross sections for the v1 = 0 and v1 = 1 vibrational levels, while rovibrational quenching from v1 = 2, j1 = 0 is a factor of ∼5 larger than that from v1 = 1, j1 = 0. Details on the PES, computed at the CCSD(T)/aug-cc-pV5Z level, and fitted with an invariant polynomial method, are also presented.