Abstract: In spite of the conceptual simplicity, the design of the free-piston Stirling engines (FPSEs) is made difficult by the necessity to accurately foresee the effect of the various geometric, dynamic, and thermodynamic variables on their behavior. This paper describes a fully developed mathematical model able to characterize these machines taking into account all of the relevant physical aspects involved, including the casing motion. A closed form solution of the governing equations is used, together with a theorem developed by the authors to derive a basic criterion for the FPSE cyclic steady operation. Because this criterion is expressed in an analytical form as function of the different variables involved, it is possible to choose the engine design parameters to ensure a steady periodic state. The developed model may be used not only for design purposes but also to simulate theoretically the dynamic behavior of a built engine.