Theoretical insights into H-abstraction by ozone from saturated species have been virtually nonexistent, in sharp contrast to the situation for reactions with unsaturated species. Our computed rate constants for reactions with tetrahydrofuran and its methyl derivatives, obtained at various levels of theory, show that abstraction occurs primarily at the carbons situated beside the heterocyclic oxygen, with least likely reaction from the methyl groups. All of the methods tested are in broad relative agreement with this conclusion and with recent experiments although they do differ widely in terms of their absolute values. Abstraction leads variously to the formation of hydrotrioxides, ROOOH, or more directly to R-OH + O-2(((1)Delta(g)). To understand some of the observed behaviors, dimethyl and diethyl acyclic ethers, methyl and ethyl hydroperoxides, and the alcohols methanol and ethanol were also included in this study. For the simplest system, CH3OH + O-3 -> H2C = O + H2O2, an all-electron scalar relativistic CCSD(T)/CBS approach yields a barrier height that differs by > 5 kJ mol(-1) when an additional multireference treatment up to CCSDT(Q) is considered.