A major subject of study in biological research is the dynamics of stem cells in squamous epithelia. Given that most common human cancers develop from epithelia, understanding the rules of cell fate decision in these systems is key to explaining not only healthy tissue growth and maintenance but also the processes of mutagenesis and cancer. The aim of my project was to investigate the dynamics in squamous epithelial tissues both in homeostasis and preneoplasia, using cellular automata (CA) models. Stem cell dynamics has been shown to be accurately described by a simple mathematical model, the single progenitor (SP) model. Reliable parameterisation of this model would give access to valuable quantitative information on epithelial tissue maintenance and enable investigating how mutations affect tissue dynamics. I initially identified the most appropriate method for accurately parameterising the homeostatic system. I then sought to account for the spatial patterning of cells by implementing the SP model in two-dimensional space. The spatial model was able to reproduce the key signatures of homeostatic dynamics, thus showing that restrictions imposed by tissue organization do not alter the neutral dynamics. Furthermore, I studied non-homeostatic dynamics in stratified squamous epithelial tissues by spatially modelling the growth and competition of non-neutral mutations as well as the effects of wounding in the tissue. The studied dynamics of Notch and p53 mutant clones in mouse epithelia has been found to be highly distinct, with the former fully colonizing the tissue whereas the latter only partially. I demonstrated that the two mutants’ tissue takeover dynamics can be recapitulated by two distinct spatial feedback rules, on the basis of response to crowding, providing a mechanistic explanation of the observed distinct growth modes. Finally, mutant competition was explored. A striking effect resulting from the spatial interaction of the two mutations in a wild-type background is that the p53 mutant cell population was always outcompeted by the Notch mutant population and appeared to shrink. Considering this consistent emergent behaviour in the competition simulations and given the paucity of Notch mutations in human cancer datasets, it is tempting to speculate that the aggressive fitness of Notch may offer a tumour-protective effect.