This talk will cover two approaches for modeling blood flow in the human body.  The first approach describes blood transport in elastic vessels and requires the numerical solution of a nonlinear hyperbolic system on branching vessel networks.  I will discuss some mathematical properties of these equations that seem to be useful for analysis of numerical schemes, and I will also describe an application of this framework to the study of blood flow in abnormal physiologies.  The second approach deals with the interaction of blood flow and heart tissue, valves, and surrounding vessels.  This approach uses a fluid/structure interaction scheme called the immersed boundary finite element method.  In this method, solid displacements and forces are approximated on a moving (Lagrangian) finite element mesh that describes biological tissue, and the blood velocity field and pressure are approximated on a fixed (Eulerian) Cartesian grid.  I will describe applications of this approach to the ongoing development of computer models for prenatal and postnatal hearts.