Dendrite crystal growth is one of the examples of crystal growth in undercooled liquids. In spite of a number of attempts to improve the widely quoted model of dendritic crystal growth from undercooled melts (Acta Metall. 35 (1987) 957) the agreement between the theory and the experiment, especially at high undercooling, is still poor. In this study, we show that the temperature of crystal/melt interface of a growing crystal into a deeply undercooled liquid can differ considerably from the melting temperature, thus making the kinetic attachment coefficient to depend on the undercooling temperature, a fact ignored by previous models. For deeply undercooled liquids we propose a new fluctuation model of atomic mobility, viscosity and diffusion, and on its basis a phenomenological model for the temperature dependence of the kinetic coefficient is developed. Analysis of the model and the experimental data shows that the slowed attachment kinetics in deeply undercooled liquids considerably decreases the velocity of crystal growth, and can even lead to glass transition. (C) 2003 Elsevier Science B.V. All rights reserved.