Averaged one-dimensional mathematical models for blood flow in compliant vessels require equations describing the dynamics of fluid flow in the form of partial differential equations (PDEs), along with a constitutive equation for the vessel wall mechanics—also known as the tube law—to account for the vessel’s compliance. The coupling of these two components results in a simple fluid-structure interaction model for blood flow in compliant (as opposed to rigid) vessels. In this chapter, we study elastic and viscoelastic tube laws, which are required to satisfy two basic criteria: First, they must accurately represent the physiological phenomena of interest. Second, when coupled with the PDEs governing blood fluid dynamics, they must preserve certain mathematical properties. Here we analyse the resulting PDEs for each tube law considered. Mathematically admissible elastic tube laws lead to hyperbolic equations. We establish their hyperbolic character and examine their complete eigenstructure, and the nature of their characteristic fields. Viscoelastic tube laws give rise to parabolic equations. We approximate these parabolic equations by hyperbolic equations with stiff source terms and analyse the resulting hyperbolised PDEs. Tube laws involve several geometric and mechanical parameters, which must typically be determined through experimentation. In this chapter, we examine both elastic and viscoelastic tube laws with constant and variable parameters, analysing the resulting PDEs. A case study of animal and human blood vessels is presented, in which a parameter estimation process is applied, and the resulting tube laws are used to replicate experimental data. It is seen that viscoelastic tube laws more accurately represent the physiology of compliant blood vessels compared to elastic tube laws and are therefore recommended for practical applications. The chapter concludes with a summary, a discussion, and a list of recommended references for further study.