In this chapter, we derive the governing equations for the dynamics of blood flow in compliant vessels. We begin by briefly reviewing the Navier-Stokes equations for compressible materials, augmented by a closure condition through an equation of state. These equations are particularly relevant to specialised medical applications, such as Traumatic Brain Injury and Extracorporeal Shock Wave Lithotripsy. The Navier-Stokes equations are then specialised for incompressible media, such as liquids at low pressure and velocity. Formulations for solving these equations are reviewed, including the artificial compressibility method and the stream function-vorticity approach. To prepare for the derivation of blood flow equations in compliant conduits, we define suitable control volumes and recall the Reynolds transport theorem. We then proceed with a detailed derivation of the one-dimensional averaged equations for the dynamics of blood flow in compliant arteries and veins. This results in a system of two first-order, time-dependent, nonlinear partial differential equations. Since these two equations contain three unknowns, rendering the system undetermined, a closure condition is added by incorporating the mechanics of the compliant vessel wall via a tube law (or pressure law). A simple example of a determined system is provided, using a basic tube law valid only for blood flow in compliant arteries. References to later chapters on generalised models, analysis, and exact solutions are included. Finally, from the established one-dimensional equations, we derive zero-dimensional models, or compartmental models, which consist of systems of ordinary differential equations in time. The chapter concludes with a summary, conclusions, and suggested references for further study.