A definition of hydrogen bonding in water is proposed in which an H...O pair forms a hydrogen bond if (a) an oxygen atom is the nearest nonchemically bonded neighbor of a hydrogen atom; and (b) the hydrogen is the first or the second intermolecular near-neighbor of the oxygen. Unlike the commonly employed hydrogen-bond definitions, this definition does not depend on the choice of geometric or energetic cutoffs applied to continuous distributions of properties. With the present definition, the distribution of O...H bond lengths decays smoothly to zero in a physically reasonable range. After correction for the presence of intermittent hydrogen bonds, this definition appears to provide a more stable description of hydrogen bonds and coordination shells than the more conventional cutoff-based definition. "Partial" H bonds satisfying only one of the two bonding requirements serve as transition states in the H-bond network evolution.