Understanding the nature of tumour evolution promises to enable more accurate prognostic methods and more effective treatment strategies. I will use three examples to illustrate how the analysis of stochastic processes can aid this goal by bridging the gap between ODE/PDE models and agent-based simulations. First, I will show how surprisingly simple mathematical expressions can be derived to explain why selective sweeps (the spread of beneficial mutations through an entire population) are rare except when tumours are relatively very small. Next, I will explain how studying tree generating processes and tree shape indices can improve model selection and clinical forecasting methods. Finally, I will present an application of stochastic processes to improving cancer cure rates by minimizing the probability of evolutionary rescue. Although all this work is motivated by questions in cancer research, the methods and results are readily applicable to other biological systems such as bacteria and invasive species.