The human cardiovascular system is a vastly complex collection of interacting components, including vessels, organ systems, valves, regulatory mechanisms, mi- crocirculations, remodeling tissue, and electrophysiological signals. Experimental, mathematical, and computational research efforts have explored various hemody- namic questions; the scope of this literature is a testament to the intricate nature of cardiovascular physiology. In this work, we focus on computational modeling of blood ow in the major vessels of the human body. We consider theoretical questions related to the numerical approximation of reduced models for blood ow, posed as nonlinear hyperbolic systems in one space dimension. Further, we apply this modeling framework to abnormal physiologies resulting from surgical intervention in patients with congenital heart defects. This thesis contains three main parts: (i) a discussion of the implementation and analysis for numerical discretizations of reduced models for blood ow, (ii) an investigation of solutions to different classes of models in the realm of smooth and discontinuous solutions, and (iii) an application of these mod- els within a multiscale framework for simulating ow in patients with hypoplastic left heart syndrome. The two numerical discretizations studied in this thesis are a characteristics{based method for approximating the Riemann{invariants of reduced blood ow models, and a discontinuous Galerkin scheme for approximating solutions to the reduced models directly. A priori error estimates are derived in particular cases for both methods. Further, two classes of hyperbolic systems for blood ow, namely the mass{momentum and the mass{velocity formulations, are systematically compared with each numerical method and physiologically relevant networks of ves- sels and boundary conditions. Lastly, closed loop vessel network models of various Fontan physiologies are constructed. Arterial and venous trees are built from net- works of one{dimensional vessels while the heart, valves, vessel junctions, and organ beds are modeled by systems of algebraic and ordinary differential equations.