A numerical model of cellular and dendritic growth has been developed which can predict cellular and dendritic growth spacings, undercoolings, interface shape and transitions between structures. An important feature of the model is that the spacing selection mechanism is treated so that no a priori assumption, such as marginal stability, is used. A small stable range of spacings is predicted, both for cells and dendrites, and these agree very well with experiment, both at low velocities and at high velocities.