In this paper, an analytic solution is provided for the stability behavior of cylindrical shells made of compositionally (or functionally) graded ceramic-metal materials under the axial compressive loads varying as a power function of time. The material properties of the compositionally graded shells are assumed to vary continuously through the thickness of the shell according to arbitrary distribution of the volume fraction of the constituents. The fundamental equations for thin shells of compositionally graded ceramic-metal material are obtained using the Love's shell theory and a solution for two initial conditions is obtained by applying Galerkin method and Lagrange-Hamilton type principle. The effects of the constituent volume fractions, the variations of loading parameters and the variations of power of time in the axial load expression on the critical parameters of the shell for two initial conditions are also elucidated. The results reveal that these effects play a major role in dictating the loss of the stability of the compositionally graded shells under the axial compressive loads. Comparing results with those in the literature validates the present analysis. (c) 2004 Published by Elsevier Ltd.