The torsional potential about the O-O single bond of dimethyl peroxide, CH3OOCH3, was investigated with the aid of large-scale ab initio calculations performed at different levels of Moller-Plesset perturbation theory and coupled-cluster expansions. Additionally, several density functional approaches were applied. For comparative purposes, the torsional potentials of methyl hydroperoxide, CH3OOH, and hydrogen peroxide, HOOH, were calculated at the same levels of approximation. In the already well-investigated case of HOOH and also for CH3OOH excellent agreement with the experimentally determined structures and barrier heights can be achieved at the coupled-cluster CCSD(T) level with the application of extended polarized basis sets augmented with diffuse functions. However, in the case of dimethyl peroxide, the peculiar shape of the computed CCSD(T)/cc-pVTZ torsional potential, with an exceedingly shallow region ranging from 110 to 250degrees, with two skew minima at about 115 and 245degrees and with a trans minimum at 180degrees, deviates significantly from that of the experimentally derived torsional potential, which has a barrier at 180degrees separating the two distinctly deeper skew minima at 120 and 240degrees. The difficulties encountered in reaching a reasonably converged result with respect to further basis set extension are discussed. It is also shown that the results of density functional theory (DFT) and Moller-Plesset second-order (MP2) calculations differ considerably from the Moller-Plesset higher-order and CCSD(T) results.