First, we revisit the stochastic Luria-Delbrück model: a classic two-type branching process which describes cell proliferation and mutation. We prove limit theorems and exact results for the mutation times, clone sizes, and number of mutants. Second, we extend the framework to consider mutations at multiple sites along the genome. The number of mutants in the two-type model characterises the mean site frequency spectrum in the multiple-sites model. Further, we provide law-of-large-numbers and large-deviations results for the infinite-sites limit. Our predictions are consistent with previously published cancer genomic data. To conclude, we briefly discuss ongoing work in collaboration with biologists on branching random walks and the phylogenetic reconstruction of cancer evolution.