The kinetics of phase transformations has been re-examined by considering the case when this is controlled by phase growth, with nearly instantaneous nucleation. In this case, the phase transformation is likely to occur at temperatures relatively close to the true thermodynamic transformation temperature T-t, and the growth rate vanishes on approaching this temperature. The actual solutions for nucleation and growth thus deviate from the classical laws proposed to describe the dependence of fraction transformed on temperature and rate of change in temperature. Similarly, the Kissinger equation becomes a poor description for the dependence of the transformation peak temperature on the rate of change in temperature. Nevertheless, the Avrami-Nakamura approach can still be used to derive formulae for these cases. Alternative solutions were thus obtained both by numerical integration and also in the form of nearly correct approximate formulae. These formulae account for sudden deviations on approaching the transition temperature. (C) 2005 Elsevier B.V. All rights reserved.