Platelet-rich blood clots (thrombi) can form within critical coronary and cerebral arteries thereby causing severe health consequences. I will discuss our ongoing efforts to model such arterial thrombosis. From underlying two-scale (vessel and platelet) models of the relevant biophysical processes, we have derived macroscale continuum models that can capture the formation of occlusive thrombi under some conditions and cycles of thrombus formation and embolization (breakup) under other conditions. Our current models are two phase mixture models in which the thrombus is a porous viscoelastic material whose extent, porosity, and viscoelastic properties evolve dynamically because of the interactions among platelets, vessel wall, and background fluid. The models account for formation and breaking of two important types of platelet-platelet bonds and for the activation of platelets necessary for some of these bonds to form. We have used the models to begin to understand how the dynamics of platelet activation, bond formation and breaking, and dynamic changes in the thrombus porosity can strongly influence the fate of a growing thrombus.