Measured heat transfer rates through turbulent and transitional boundary layers on an isothermal, convexly curved wall show Stanton numbers 20-50 percent below flat wall values. Recovery is slow on aflat wall downstream of the curve; after 60 cm, Stanton numbers were 15-20 percent below flat wall values. Five secondary effects were studied: (i) initial boundary layer thickness, (ii) free-stream velocity, (iii) free-stream acceleration, (iv) unhealed starting length, and (v) transition. Regardless of the initial state, curvature without acceleration eventually forced the boundary layer into an asymptotic condition: (formula presented). Strong acceleration with curvature brought the exponent on (formula presented).