In a thermally bistable medium, cold, dense gas is separated from warm, rarified gas by thin phase transition layers, or fronts, in which radiative heating/cooling, thermal conduction, and convection of material are balanced. While these fronts have received only scant attention in the literature, and are not resolved by most current numerical simulations, they have been shown to have important ramifications for transport processes and structure formation in the diffuse interstellar medium. Here, we discuss calculations of their hydrodynamic and magnetohydrodynamic stability properties.