It is proposed that the benzene dimer has two almost isoenergetic structures. This has been confirmed by very high level ab initio calculations. MP2, MP4, and CCSD(T) calculations with various basis sets up to the augmented correlation consistent polarized double-zeta (aug-cc-pvdz) one were performed for the sandwich (S), T-shaped (T), and parallel-displaced (PD) structures of the benzene dimer. MP2 strongly overestimates the stabilization energy and leads to incorrect global minimum. Also, the MP4 predictions differ from the CCSD(T) ones. At the CCSD(T) level the T structure was the most stable structure with stabilization energy of about 2.3 kcal/mol. The PD structure at the same level was found to be slightly less stable (by 0.2-0.3 kcal/mol). The energy difference between these structures is sensitive to the theoretical level applied, and we estimated that both structures are similarly stable. The absolute value of stabilization energy is believed to be close to the genuine value; this conclusion was based on study of basis set saturation where basis sets up to [4s3p2d1f] were utilized. Performance of existing empirical potentials is evaluated. The existence of the C-H ...pi H-bond was not confirmed. Finally, consequences of the existence of two similarly stable structures of the benzene dimer for the structure and function of proteins are discussed.