The failure mode and strength of rock are often evaluated using failure criteria that (i) neglect the intermediate principal stress or (ii) examine conditions over a limited range ofmean stress. Review of the literature, however, shows that tests involving multi-axial stress states suggest that all three principal stresses should be considered in evaluating yield or failure. Further, rock displays a pressure dependence that can be interpreted as a change in friction with mean stress. To explore the nature of stress states at failure, experiments and published data were analyzed for Dunnville sandstone within the framework of stress invariants p, q, and µ, where p =mean stress, q = deviatoric stress, and µ = Lode angle. A series of conventional triaxial compression, conventional triaxial extension, and true-triaxial tests were conducted. Published data for Dunnville sandstone were collected and a database consisting of similar tests on Dunnville sandstone was developed. The axisymmetric compression data were fitted to three failure criteria: Mohr-Coulomb (MC), Hoek-Brown (HB), Paul-Mohr-Coulomb (PMC), a generalized linear criterion containing all three principal stresses. This criterion was evaluated using three parameter and six parameter formulations. The results show that the six-parameter PMC provides the best approximation of the test data and successfully captures, in a piecewise-linear manner, the well-known nonlinear nature of the failure surface. The Paul-Mohr-Coulomb criterion representation in the (p ¡q) diagram involves a failure surface in principal stress space that can be described as a 6-12-6 sided pyramid. The thesis presents analyses, including data from multi-axial stress states, and discussion on the three failure criteria, including details of the generalized linear condition as well as a simplified version of Paul-Mohr-Coulomb failure criterion.