Two modifications of Gaussian-4 (G4) theory [L. A. Curtiss et al., J. Chem. Phys. 126, 084108 (2007)] are presented in which second- and third-order perturbation theories are used in place of fourth-order perturbation theory. These two new methods are referred to as G4(MP2) and G4(MP3), respectively. Both methods have been assessed on the G3/05 test set of accurate experimental data. The average absolute deviation from experiment for the 454 energies in this test set is 1.04 kcalmol for G4(MP2) theory and 1.03 kcalmol for G4(MP3) theory compared to 0.83 kcalmol for G4 theory. G4(MP2) is slightly more accurate for enthalpies of formation than G4(MP3) (0.99 versus 1.04 kcalmol), while G4(MP3) is more accurate for ionization potentials and electron affinities. Overall, the G4(MP2) method provides an accurate and economical method for thermochemical predictions. It has an overall accuracy for the G3/05 test set that is much better than G3(MP2) theory (1.04 versus 1.39 kcalmol) and even better than G3 theory (1.04 versus 1.13 kcalmol). In addition, G4(MP2) does better for challenging hypervalent systems such as H(2)SO(4) and for nonhydrogen species than G3(MP2) theory.