Somatic mutation plays a key role in transforming normal cells into cancerous cells. The analysis of cancer progression therefore requires the study of how point mutations and chromosomal mutations accumulate in cellular lineages. The spread of somatic mutations depends on the mutation rate, the number of cell divisions in the history of a cellular lineage, and the nature of competition between different cellular lineages. We consider how various aspects of tissue architecture and cellular competition affect the pace of mutation accumulation. We also discuss the rise and fall of somatic mutation rates during cancer progression. BioEssays 26:291–299, 2004. © 2004 Wiley Periodicals, Inc.