Turnbull’s transformation nucleus model initiated the concept of a growth barrier for a spherical-cap crystal nucleus growing on a small flat substrate. The recently developed free growth model provided a clear physical picture of the growth barrier concept with experimental and modelling support. Fletcher’s spherical substrate model enhanced the understanding of the geometrical effect of a substrate on nucleation. A recent novel analysis of Fletcher’s model furnished new insights into the similarities and differences between nucleation on spherical and flat substrates. It is necessary to distinguish between the undercooling required for nucleation and that for overcoming the growth barrier; the greater one determines the early stages of grain formation.