Measurements of the heat transfer by natural convection across vertical and inclined air layers are reported. The air layer is bounded by two parallel isothermal flat plates and around the edges by a low thermal conductivity wall of thickness B. A critical wall thickness Bc was determined by solving the two-dimensional conduction equation in the bounding wall region: for B>Bc the convective heat transfer should be insensitive to B. Measurements are reported for air layers with B>Bc and an aspect ratio of 5. They cover a range in Rayleigh number from 103 to 108 , and a range in orientation from horizontal to vertical. The effect of the emissivity of the bounding wall on the heat transfer across the air layer was evaluated from measurements obtained respectively with a low and a high value of the wall emissivity. The paper proposes terminology required to define the thermal conditions, and discusses the impact of these boundary conditions on the total heat transfer.