This paper gives a critical review of the various theories for predicting the maximum pressure that have been applied to thick-walled cylindrical pressure vessels. A new anisotropic theory proposed by the authors is also reviewed, and all the available experimental results are compared with this and other theories. Based upon agreement with test results and relative simplicity of application, the method developed by the authors is recommended. In recommending the proposed theory, it should be noted that the study is restricted to materials with instability loads, as characterized by a horizontal slope in the tensile nominal stress-strain diagram.