A recently computed, high-accuracy ab initio Born-Oppenheimer (BO) potential energy surface (PES) for the water molecule is combined with relativistic, adiabatic, quantum electrodynamics, and, crucially, nonadiabatic corrections. Calculations of ro-vibrational levels are presented for several water isotopologues and shown to have unprecedented accuracy. A purely ab initio calculation reproduces some 200 known band origins associated with seven isotopologues of water with a standard deviation (sigma) of about 0.35 cm(-1). Introducing three semiempirical scaling parameters, two affecting the BO PES and one controlling nonadiabatic effects, reduces sigma below 0.1 cm(-1). Introducing one further rotational nonadiabatic parameter gives a better than 0.1 cm(-1) for all observed ro-vibrational energy levels up to J = 25. We conjecture that the energy levels of closed shell molecules with roughly the same number of electrons as water, such as NH3, CH4, and H3O+, could be calculated to this accuracy using an analogous procedure. This means that near-ab initio calculations are capable of predicting transition frequencies with an accuracy only about a factor of 5 worse than high resolution experiments.