Abstract A recent numerical model of cellular and dendritic growth has been extended into the high velocity region where the distribution coefficient, liquidus slope and diffusion coefficient depend on the growth velocity. The primary spacing selection mechanism is modelled so that no a priori assumptions need be made about a spacing selection condition. The results are compared with experimental primary spacing measurements obtained using rapid laser resolidification and good agreement is found. The numerical results for undercooling and tip radii are compared with those predicted for dendrites using marginal stability arguments, showing the potential and limits of the analytical models. The effect of high velocity on microsegregation is examined and microsegregation profiles are predicted.