This chapter focuses on the mathematical theory of nonlinear hyperbolic equations, in which discontinuities can form over time from smooth initial conditions. We begin with scalar nonlinear equations, studying rarefaction and shock waves. Emphasis is placed on shock formation from continuous initial data, which renders invalid the differential form of the equation, as derivatives may no longer be defined. Generalised solutions, which accommodate both continuous and discontinuous solutions, arise from introducing the integral form of the conservation laws. These solutions now satisfy the differential form in smooth regions and the Rankine-Hugoniot condition at discontinuities. However, the enlarged set of solutions also includes physically non-admissible, spurious solutions. Examples of non-uniqueness are provided, and a selection criterion—the Lax entropy condition—is introduced to eliminate unwanted solutions. The Riemann problem for the Burgers equation is then introduced and solved. Systems of hyperbolic equations are discussed, along with definitions and examples. Eigenvalues, eigenvectors, and the notion of hyperbolicity are introduced, as well as characteristic fields and their properties. The integral form for systems of hyperbolic equations is presented, along with the corresponding Rankine-Hugoniot conditions. Nonlinear blood flow equations are introduced as a relevant example of a system of hyperbolic equations. The mathematical concepts studied are applied to analyse a simplified version of the blood flow equations, assuming the Coriolis coefficient to be \(\alpha =1\) . These equations are also expressed in terms of mathematical conservation laws, but using physical variables rather than conserved variables. A cautionary note is provided to ensure that the reader correctly interprets such equations. The chapter concludes with a summary and recommendations for further study, along with a list of references. A selection of exercises is included at the end of the chapter.