The significant structure theory (SST) for liquid viscosities, originally proposed by Eyring, coupled with a cubic equation of state was used for the simultaneous correlation of gas and liquid viscosities of pure fluids (polar and nonpolar) at saturated conditions. The SST visualizes a liquid as having both "solidlike" and "gaslike" degrees of freedom with "fluidized vacancies" of molecular size randomly distributed throughout a quasi-lattice structure. In this context, the viscosity of a liquid is calculated from two main components: a gaslike eta(g) and a solidlike eta(s) contribution. The first viscosity contribution eta(g) represents the viscosity of a pure fluid at dilute gas conditions (low-pressure viscosity). The method of Chung et al. based on the Chapman-Enskog kinetic theory of gases was used to calculate eta(g). The second contribution eta(s) captures the solidlike effects on viscosity. This quantity was calculated by means of Eyring's absolute rate theory. All the thermodynamic properties required in the viscosity model were computed via the use of a well-known cubic equation of state (Soave-Redlich-Kwong or Peng-Robinson) thus allowing the simultaneous correlation of gas-liquid viscosities along their coexistence curve. The resulting model was satisfactorily validated in the representation of experimental saturated gas and liquid viscosities of a highly polar compound (water) and a nonpolar fluid (propane) over a wide range of temperatures (from near the triple point up to the critical region of the fluid of interest).