The transport of pure gases and of binary gas mixtures through a microporous composite membrane is discussed. The membrane consists of an alumina support with a mean pore diameter of 160 nm and an alumina top (separation) layer with pores of 2-4 nm. The theory of Knudsen diffusion, laminar flow and surface diffusion is used to describe the transport mechanisms. It appears for the composite membrane that Knudsen diffusion occurs in the toplayer and combined Knudsen diffusion/laminar flow in the support at pressure levels lower than 200 kPa. For the inert gas mixture H2/N2 separation factors near 3 could be achieved which is 80% of the theoretical Knudsen separation factor. This value is shown to be the product of the separation factor of the support (1.9) and of the top layer (1.5). The value for the top layer is rather low due to the relatively small pressure drop across this layer. This situation can be improved by using composite membranes consisting of three or more layers resulting in a larger pressure drop across the separation layer. CO2 surface diffusion occurs on the internal surface of the investigated alumina membranes. At 250-300 K and a pressure of 100 kPa the contribution of surface diffusion flow measured by counterdiffusion is of the same order of magnitude as that resulting from gas diffusion. The adsorption energy amounts —25 kJ/mol and the surface coverage is 20% of a monolayer at 293 K and 100 kPa. The calculated surface diffusion coefficient is estimated to be 2-5 x 10-9 m2/sec. Modification of the internal pore surface with MgO increases the amount of adsorbed CO2 by 50-100%. Modifications with finely dispersed silver are performed to achieve O2 surface diffusion.