Geometries, anharmonic vibrations, and torsion-wagging (TW) multiplets of hydrazine and its deuterated species are studied using high-level ab initio methods employing the second-order Mo̸ller-Plesset perturbation theory (MP2) as well as the coupled cluster singles and doubles model including connected triple corrections, CCSD(T), in conjunction with extended basis sets containing diffuse and core functions. To describe the splitting patterns caused by tunneling in TW states, the 3D potential energy surface (PES) for the large-amplitude TW modes is constructed. Stationary points in the 3D PES, including equivalent local minima and saddle points are characterized. Using this 3D PES, a flexible Hamiltonian is built numerically and then employed to solve the vibrational problem for TW coupled motion. The calculated ground state r(av) structure is expected to be more reliable than the experimental one that has been determined using a simplified structural model. The calculated fundamental frequencies allowed resolution of the assignment problems discussed earlier in the literature. The determined energy barriers, including the contributions from the small-amplitude vibrations, to the tunneling of the symmetric and antisymmetric wagging mode of 1997 cm(-1) and 3454 cm(-1), respectively, are in reasonable agreement with the empirical estimates of 2072 cm(-1) and 3312 cm(-1), respectively [W. Łodyga et al. J. Mol. Spectrosc. 183, 374 (1997)]. However, the empirical torsion barrier of 934 cm(-1) appears to be overestimated. The ab initio calculations yield two torsion barriers: cis and trans of 744 cm(-1) and 2706 cm(-1), respectively. The multiplets of the excited torsion states are predicted from the refined 3D PES.