Functionally-graded materials (FGM), are those materials in which some specific physical properties change gradually with respect to material dimensions. FGM have recently been widely used. Between the different types of FGM, this book is interested in studying the stress analysis of the Chemical Composition Gradient FGM, in which structure composed of two dissimilar material constituents that are gradually changing from one side to the other side. This change is considered herein to vary according to an exponential function. the aim of this book is to derive mathematical formulas that can be used to determine the mechanical and thermal stresses induced in an FGM cylindrical vessel for which the material properties vary according to an exponential function and subjected to mechanical and thermal loads. This solution is achieved with the help of the world's leading computational software package "Mathematica" and as a verification step a thermo-mechanical coupled FE model is constructed and solved with the help of the global leader software in engineering simulation "Ansys".