The author proves the KAM twist-mapping theorem using renormalisation group methods. Besides being of great independent interest, this approach seems to work for far greater perturbations from integrable than the classical KAM theory. It also provides an instructive method to construct the invariant curves in an explicit way which can be numerically exploited. The author also gets smoothness on the invariant curves, the degree of which depends on the smoothness of the map and the Liouville exponent of the rotation number.