The intertwined processes of platelet deposition and coagulation can lead to the development of blood clots inside a blood vessel or on an implanted medical device. Their disregulation is responsible for immense morbidity and mortality, particularly in the western world. The development of a blood clot involves complex interactions of diverse type (e.g., biochemistry, cell signaling and adhesion, fluid dynamics) on diverse spatial and temporal scales. Exploring how these interactions function together as a system is a task for which mathematical/computational modeling is well suited. In this tutorial talk, I will give a brief introduction to the biology, chemistry, and physics of blood clotting, and I will describe two of our recent efforts to model different aspects of clotting. Both models include the formation of platelet masses (thrombi) adherent to the vascular wall. In one model the developing mass is treated as a porous material whose porosity evolves with the addition of platelets to the thrombus. This model includes a comprehensive treatement of the coagulation system and its interaction with platelet deposition. The other model emphasizes the evolution of the thrombus' mechanical properties as it forms (and sometimes breaks up) in situations in which fluid-mediated stresses vary substantially.