There is a growing appreciation for the insight mathematical models can yield on biological systems. In particular, due to the challenges inherent in experimental observation of disease progression, models describing the genesis, growth and evolution of cancer have been developed. Many of these models possess the common feature that one particular type of cellular population initiates a further, distinct population. This thesis explores two models containing this feature, which also employ branching processes to describe population growth. Firstly, we consider a deterministically growing wild type population which seeds stochastically developing mutant clones. This generalises the classic Luria- Delbruck model of bacterial evolution. We focus on how differing wild type growth manifests itself in the distribution of clone sizes. In our main result we prove that for a large class of wild type growth, the long-time limit of the clone size distribution has a general two-parameter form, whose tail decays as a power-law. In the second model, we consider a fully stochastic system of cells in a growing population that can undergo birth, death and transitions. New cellular types appear via transitions, examples of which are genetic mutations or migrations bringing cells into a new environment. We concentrate on the scenario where the original cell type has the largest net growth rate, which is relevant for modelling drug resistance, due to fitness costs of resistance, or cells migrating into contact with a toxin. Two questions are considered in our main results. First, how long do we wait until a cell with a specific target type, an arbitrary number of transitions from the original population, exists. Second, which particular sequence of transitions initiated the target population. In the limit of small final transition rates, simple, explicit formulas are given to answer these questions.