We investigate models for the growth of crystals of monodisperse polymer chains. A transition from extended chain crystals to folded chain crystals occurs as the temperature is changed which is governed by the kinetics of adsorption and desorption of chains at the growth surface. Folded chains may build up on the surface of growing extended chain crystals and slow down the growth. This effect is particularly strong at the transition temperature, where the folded chains are marginally stable. Hence there is a sharp minimum in the growth velocity at this point. We compare the results of the model to experiments on crystallization of long alkanes and poly (ethylene oxide) fractions. In some cases there is an instability in the growth surface leading to patterns of crevices resembling those seen during the thickening of folded chain alkane lamellae. Sliding and rearrangement of molecules underneath the crystal surface is important to allow smooth surface growth. Sliding processes are also the dominant factor controlling the growth rate at temperatures close to the transition.