Water 2H and 17O spin relaxation is used to study water dynamics in the hydration layers of two small peptides, two globular proteins and in living cells of two microorganisms. The dynamical heterogeneity of hydration water is characterized by performing relaxation measurements over a wide temperature range, extending deeply into the supercooled regime, or by covering a wide frequency range. Protein hydration layers can be described by a power-law distribution of rotational correlation times with an exponent close to 2. This distribution comprises a small fraction of protein-specific hydration sites, where water rotation is strongly retarded, and a dominant fraction of generic hydration sites, where water rotation is as fast as in the hydration shells of small peptides. The generic dynamic perturbation factor is less than 2 at room temperature and exhibits a maximum near 260 K. The dynamic perturbation is induced by H-bond constraints that interfere with the cooperative mechanism that facilitates rotation in bulk water. Because these constraints are temperature-independent, hydration water does not follow the super-Arrhenius temperature dependence of bulk water. Water in living cells behaves as expected from studies of simpler model systems, the only difference being a larger fraction of secluded (strongly perturbed) hydration sites associated with the supramolecular organization in the cell. Intracellular water that is not in direct contact with biopolymers has essentially the same dynamics as bulk water. There is no significant difference in cell water dynamics between mesophilic and halophilic organisms, despite the high K+ and Na+ concentrations in the latter.