The stability of a closed two-phase thermosyphon model is studied as a function of the heat input, liquid filling, viscosity, density, capillary number and two-phase flow Grashof number. It is shown that for small liquid fillings or pool heights, the liquid dries out and there are no solutions beyond this point. For relatively large liquid pool heights, the heat input, when plotted versus the respective steady-state vapour height, is approximately a quadratic expression such that there are two possible steady states for each heat input. A linear stability analysis indicates that there is a turning point at the maximum heat input at which one of the eigenvalues of the linear stability analysis becomes zero. Control parameter surfaces have been calculated in order to determine the effects of the thermosyphon length, liquid pool height, capillary number, liquid-to-vapour density and viscosity ratios, heat input, and two-phase flow Grashof number on the stability of the closed two-phase thermosyphon model. It has been found that the steady operation of a thermosyphon is insensitive to the vapour-to-liquid viscosity ratio and thermosyphon length, and very sensitive to the vapour-to-liquid density ratio. Linear and nonlinear transient analyses of the thermosyphon behaviour near the steady-state operating conditions indicate that, for large liquid fillings, there are upper limits for the heat input. Large liquid fillings may result in an unstable operation of the thermosyphon. It is also shown that a linear analysis underestimates grossly the transient behaviour of the thermosyphon operation.