Hemostasis is the normal, healthy process in which a blood clot forms to stop bleeding in the event of an injury. Blood clot formation is a complex and nonlinear process that occurs under flow and on multiple spatial and temporal scales. Defects and perturbations in the hemostatic system can result in serious bleeding or pathological clot formation, but due to the complexity of the system, the responses to these changes and the underlying mechanisms are challenging to predict. Mechanistic mathematical models of blood clot formation and coagulation can elucidate biochemical and biophysical mechanisms, help interpret experimental data, and guide experimental design. In this talk I will briefly describe such models and show how our integrated mathematical and experimental approach has facilitated the discovery of previously unrecognized interactions within the clotting system.