The physical shape and structure of plants are manifestations of the actions of gene products and their concerted responses to their environment. In this chapter we introduce the plant cortical microtubule array. This structure is both a nexus in the control of plant cell shape and function, and a fascinating out-of-equilibrium system for state-of-the-art physics research. We describe how analytical and computational approaches complement each other in the study of the array, and highlight some recent results and open research questions.