Tumourigenesis occurs when a series of genome alterations occur in the same group of cells and cause uncontrolled cell proliferation. Therefore, to understand the journey from healthy to cancerous tissue, it is important to study the accumulation and spread of mutations in pre- cancerous normal tissues. Recent studies have shown that apparently normal epithelium contains a high burden of mutations in cancer-associated genes. This thesis explores the behaviour of mutant clones in normal epithelium and how this affects cancer development. The nature of mutant clonal growth and competition in normal epidermis has been a subject of debate. A study found that mutant clone sizes inferred from DNA sequencing of normal human eyelid skin were consistent with a mathematical model of neutral cell dynamics, appearing to contradict a genetic analysis of the same dataset that found several genes under positive selection. I investigate this debate using computational modelling that takes into account the tissue structure and experimental tissue-sampling methods. The results show that mutant clone sizes in skin and oesophagus are consistent with non-neutral clonal competition. Further evidence for non-neutral selection in normal epithelium is found in patterns of mutations detected by DNA sequencing. By adapting a statistical method used for driver gene discovery, I look for enrichment or depletion of structural categories of missense mutations and find biologically meaningful patterns of selection in several proteins. The method can associate changes to protein structure or function with cell fitness, even in the absence of hotspot mutations and in the presence of passenger mutations. I demonstrate how the method is flexible and could be widely applicable, but can also produce misleading results if confounding sources of selection are not accounted for. Clonal competition in normal oesophageal epithelium is dominated by Notch1 loss-of- function mutations. I fit stochastic models of clonal dynamics to lineage tracing data to show that haploinsufficiency greatly accelerates Notch1 mutant expansion and that the loss of the second Notch1 allele provides a further strong selective advantage, consistent with the high frequency of NOTCH1 loss-of-heterozygosity events observed in human oesophagus. Finally, I examine a consequence of the spread of these highly fit mutant clones in the normal tissue. I use a mathematical model to analyse the results of a series of experiments in mutagen-treated mouse oesophagus, finding that microscopic tumours can be eliminated by highly fit clones in the surrounding normal tissue.