We have investigated the nuclear spin relaxation properties of (1)H in liquid water with the help of molecular dynamics simulations. We have computed the (1)H nuclear spin relaxation times T1 and T2 and determined the contribution of the different interactions to the relaxation at different temperatures and for different classical water models (SPC/E, TIP3P, TIP4P, and TIP4P/2005). Among the water models considered, the TIP4P/2005 model exhibits the best agreement with the experiment. The same analysis was performed with Car-Parrinello ab initio molecular dynamics simulations of bulk water at T = 330 K, which provided results close to the experimental values at room temperature. To complete the study, we have successfully accounted for the temperature-dependence of T1 and T2 in terms of a simplified model, which considers the reorientation in finite angle jumps and the diffusive translation of water molecules.