Structural, density, entropy, and diffusivity anomalies of the TIP4P/2005 model of water are mapped out over a wide range of densities and temperatures. The locus of temperatures of maximum density (TMD) for this model is very close to the experimental TMD locus for temperatures between 250 and 275 K. Four different water models (mTIP3P, TIP4P, TIP5P, and SPC/E) are compared with the TIP4P/2005 model in terms of their anomalous behavior. For all the water models, the density regimes for anomalous behavior are bounded by a low-density limit at around 0.85-0.90 g cm(-3) and a high-density limit at about 1.10-1.15 g cm(-3). The onset temperatures of the density anomaly in the various models show a much greater variation, ranging from 202 K for mTIP3P to 289 K for TIP5P. The order maps for the various water models are qualitatively very similar with the structurally anomalous regions almost superimposable in the q(tet)-τ plane. Comparison of the phase diagrams of water models with the region of liquid-state anomalies shows that the crystalline phases are much more sensitive to the choice of water models than the liquid state anomalies; for example, SPC/E and TIP4P/2005 show qualitatively similar liquid state anomalies but very different phase diagrams. The anomalies in the liquid in all the models occur at much lower pressures than those at which the melting line changes from negative to positive slope. The results in this study demonstrate several aspects of structure-entropy-diffusivity relationships of water models that can be compared with experiment and used to develop better atomistic and coarse-grained models for water.