The conrotatory and disrotatory mechanisms of the isomerization of bicyclo[1.1.0]butane to trans-buta-1,3-diene have been computationally investigated with the CASSCF, MCQDPT2, (U)B3LYP, CCSD(T), CR-CCSD(T), and CR-CC(2,3) approaches. The coupled-cluster (CC) methods, including the CC approach with singles, doubles, and noniterative triples (CCSD(T)), and its completely renormalized (CR) extensions called CR-CCSD(T) and CR-CC(2,3), and the density functional theory B3LYP approach do an excellent job of correctly predicting the activation barrier for the conrotatory pathway, which corresponds to a weakly biradical transition state (TS), producing values within experimental error bars. In particular, the recently developed CR-CC(2,3) method gives 40.8 or 41.1 kcal/mol, in perfect agreement with the experimental value of 40.6 +/- 2.5 kcal/mol. The complete-active-space self-consistent-field (CASSCF) approach and the second-order multireference perturbation theory (MCQDPT2) are less accurate in describing the conrotatory barrier than CR-CC(2,3). The higher energy disrotatory pathway, which has not been characterized experimentally and which involves a strongly biradical TS, poses a great challenge for many methods. CCSD(T) fails, predicting the activation barrier for the disrotatory pathway significantly below the conrotatory barrier, contradicting the experimental result that the conrotatory pathway describes the mechanism. The strongly biradical character of the disrotatory TS, spin contamination, and the proximity of singlet and triplet potential energy surfaces cause difficulties for B3LYP, which does not link this TS with gauche-buta-1,3-diene. No such difficulties occur in the CASSCF calculations, which offer a proper description of the structure of the disrotatory TS that links it with the reactant and product molecules. The CR-CC(2,3) approach, which accurately balances dynamical and nondynamical correlations in systems containing closed-shell and biradical structures, predicts the activation enthalpy for the disrotatory mechanism of similar to 66 kcal/mol. CR-CCSD(T) gives similar to 69 kcal/mol. In agreement with experiment and earlier multireference configuration interaction calculations of Nguyen and Gordon, CR-CCSD(T) and CR-CC(2,3) favor the conrotatory mechanism. The CASSCF, MCQDPT2, and B3LYP methods correctly place the disrotatory barrier above the conrotatory one, but, on the basis of a comparison with the accurate CR-CC(2,3) results, they underestimate the activation energy for the disrotatory pathway. All CC approaches employed in this study produce very good estimates of the enthalpy of isomerization of bicyclo[1.1.0]butane into buta-1,3-diene, the experimental value of which is -25.9 +/- 0.4 kcal/mol, giving about -28 kcal/mol, when trans-buta-1,3-diene is used as a product, and -25 kcal/mol, when the nearly isoenergetic gauche-buta-1,3-diene rotamer is used as a product. The CC reaction enthalpies are more accurate than those obtained with CASSCF, MCQDPT2, and B3LYP.