The effect of curvature on detonation speed and structure for detonation waves in C-O is investigated. Weakly curved detonation fronts have a sonic point inside the reaction zone. In such waves the detonation speed depends on the detailed internal structure and not on simple jump conditions. Hence, in order to obtain the correct propagation speed and products of burning, the reaction length-scales must be resolved in any numerical simulation involving curved detonations in C-O. For each value of the initial density there is a corresponding extinction curvature above which quasi-steady detonations cannot propagate. For densities less than 2 x 10(7) g cm(-3), where the self-sustaining planar waves are Chapman-Jouguet, and for realistic values of the curvature. the sonic paint moves from the end of silicon burning to the end of oxygen burning. Hence the effective detonation length, i.e, the length-scale of the burning between the shock and the sonic point which can affect the front, is several orders of magnitudes less than the planar waves predict. However, silicon burning, which occurs downstream of the sonic point, is increased in length by a few orders of magnitude owing to lower detonation speeds and temperatures. Therefore more intermediate-mass elements will be produced by incomplete burning if curvature is taken into account, Recent advances in detonation theory and modelling are also discussed in the context of Type Ia supernovae.