An analysis is presented of the heat transfer mechanisms operating in TBCs used in gas turbine engines. The three main mechanisms involved are: 1) Phonon conduction through the solid ceramic regions 2) Knudsen collisions of gas molecules with the walls of fine pores 3) Radiation transfer through a porous ceramic structure, with multiple scattering. It is difficult to create laboratory conditions close to those experienced during service (high temperatures, up to about 1400?C at the free surface, very high thermal gradients, up to around 400?C mm-1 in the top coat, and relatively high pressures, up to about 40 atmospheres). The development of reliable models is therefore important if the effects of service conditions on the thermal performance of the TBC are to be properly understood. An Eshelby-based method was employed, in which the pores are represented as ellipsoids. Comparisons between predicted and measured conductivity data in general show good agreement. The radiative contribution becomes significant at around 1500 K. At a given temperature, it is predicted to be more significant in PVD TBCs than in plasma sprayed coatings. Furthermore, the radiative contribution rises as sintering occurs during service, leading to healing of certain defects and a consequent increase in the photon mean free path. Knudsen conduction through the gas contained in pores becomes more significant as the gas pressure is raised and it is predicted that the effective conductivity under service pressures will be appreciably higher than that at the ambient pressures used in obtaining most experimental data. It is concluded that laboratory measurements of the effective conductivity are likely to underestimate the true value under service conditions by a factor of up to around 50%.