We use the Mayer sampling method, with both direct and overlap sampling, to calculate and compare classical virial coefficients up to B-6 for various water models (SPC, SPC/E, MSPC/E, TIP3P, and TIP4P). The precision of the computed values ranges from 0.1% for B-2 to an average of 25% for B6- When expressed in a form scaled by the critical properties, the values of the coefficients for SPC water are observed to greatly exceed the magnitude of corresponding coefficients for the simple Lennard-Jones model. We examine the coefficients in the context of the equation of state and the Joule-Thomson coefficient. Comparisons of these properties are made both to established molecular simulation data for each respective model and to real water. For all models, the virial series up to B-5 describes the equation of state along the saturated vapor line better than the series that includes B-6. At supercritical temperatures, however, the sixth-order series often describes pressure-volume-temperature behavior better than the fifth-order series. For example, the sixth-order virial equation of state for SPC/E water predicts the 673 K isotherm within 8% of published molecular simulation values up to a density of 9 mol/L (roughly half the critical density of SPC/E water).