Summary: We present a theoretical, high Reynolds number, investigation into the combined influence of surface cooling and curvature on the stability and transition properties of a compressible axisymmetric boundary layer. We consider the boundary-layer flow along a semi-infinite circular cylinder, where the temperature of the cylinder surface is kept significantly below the free-stream temperature, and the cylinder radius is comparable with the boundary-layer thickness. We concentrate on linearviscous-inviscid interactive Tollmien-Schlichting modes, rather than on the inviscid inflectional kind.