Mathematical models and simulation studies are powerful tools to investigate dynamic properties of complex systems. Specifically, they can be used to test alternative hypotheses on underlying biological mechanisms for their consistency with real data and therefore to effectively guide the design of new experimental strategies or clinical trials. In this study, we present an overview of recently published mathematical approaches applied to the description of chronic myeloid leukemia (CML). We discuss three different fields relevant to clinical issues: the pathogenesis of the malignancy, the treatment effects of the tyrosine kinase inhibitor imatinib, and the process of acquired treatment resistance highlighting both the differences and the consistencies in the proposed hypotheses and the resulting conclusions. The mathematical models presented agree that CML can adequately be described as a clonal competition between normal and leukemic stem cells for a common resource. Furthermore, a certain therapeutic effect of imatinib on leukemic stem cells turned out to be necessary to consistently explain clinical data on the long-term response of CML patients under imatinib treatment. However, the approaches described cannot resolve the question whether or not this effect is sufficient to ultimately eradicate malignant stem cells. A number of different hypotheses have been proposed concerning the initiation and the dynamics of treatment-resistant malignant stem cell clones. The theoretical results clearly indicate that further experimental effort with the particular focus on the quantitative monitoring of resistant clones will be required to definitely distinguish between these hypotheses.