Accurate barriers for the 1,3-dipolar cycloadditions of ozone with acetylene and ethylene have been determined via the systematic extrapolation of ab initio energies within the focal point approach of Allen and co-workers. Electron correlation has been accounted for primarily via coupled cluster theory, including single, double, and triple excitations, as well as a perturbative treatment of connected quadruple excitations [CCSD, CCSD(T), CCSDT, and CCSDT(Q)]. For the concerted [4 + 2] cycloadditions, the final recommended barriers are Delta H-(0K)(double dagger) = 9.4 +/- 0.2 and 5.3 +/- 0.2 kcal mol(-1) for ozone adding to acetylene and ethylene, respectively. These agree with recent results of Cremer et al. and Anglada et al., respectively. The reaction energy for 03 + C2H2 exhibits a protracted convergence with respect to inclusion of electron correlation, with the CCSDT/cc-pVDZ and CCSDT(Q)/cc-pVDZ values differing by 2.3 kcal mol(-1). Recommended enthalpies of formation (298 K) for cycloadducts 1,2,3-trioxole and 1,2,3-trioxolane are +32.8 and -1.6 kcal mol(-1), respectively. Popular composite ab initio approaches [CBS-QB3, CBS-APNO, G3, G3133, G3(MP2)B3, G4, G4(MP3), and G4(MP2)] predict a range of barrier heights for these systems. The CBS-QB3 computed barrier for ozone and acetylene, Delta H-(0K)(double dagger) = 4.4 kcal mol(-1), deviates by 5 kcal mol(-1) from the focal point value. CBS-QB3 similarly underestimates the barrier for the reaction of ozone and ethylene, yielding a prediction of only 0.7 kcal mol(-1). The errors in the CBS-QB3 results are significantly larger than mean errors observed in application to the G2 test set. The problem is traced to the nontransferability of MP2 basis set effects in the case of these reaction barriers. The recently published G4 and G4(MP2) approaches perform substantially better for 03 + C2H2, predicting enthalpy barriers of 9.0 and 8.4 kcal mol(-1), respectively. For the prediction of these reaction barriers, the additive corrections applied in the majority of the composite approaches considered lead to worse agreement with the reference focal point values than would be obtained relying only on single point energies evaluated at the highest level of theory utilized within each composite method.