Auxin and cytokinin are key hormonal signals that control the cellular architecture of the primary root and the initiation of new lateral root organs in the plant Arabidopsis thaliana. Both developmental processes are regulated by cross-talk between these hormones and their signalling pathways. In this paper, sub-cellular and multi-cellular mathematical models are developed to investigate how interactions between auxin and cytokinin influence the size and location of regions of division and differentiation within the primary root, and describe how their cross-regulation may cause periodic branching of lateral roots. We show how their joint activity may influence tissue-specific oscillations in gene expression, as shown in Moreno-Risueno et al. (2010) and commented upon in Traas and Vernoux (2010), and we propose mechanisms that may generate synchronisation of such periodic behaviours inside a cell and with its neighbours. Using a multi-cellular model, we also analyse the roles of cytokinin and auxin in specifying the three main regions of the primary root (elongation, transition and division zones), our simulation results being in good agreement with independent experimental observations. We then use our model to generate testable predictions concerning the effect of varying the concentrations of the auxin efflux transporters on the sizes of the different root regions. In particular, we predict that over-expression of the transporters will generate a longer root with a longer elongation zone and a smaller division zone than that of a wild type root. This root will contain fewer cells than its wild type counterpart. We conclude that our model can provide a useful tool for investigating the response of cell division and elongation to perturbations in hormonal signalling. © 2012 Elsevier Ltd.