This chapter focuses on the basic mathematical theory of linear constant-coefficient hyperbolic partial differential equations, whose solutions depend on both space and time. We begin by introducing the initial-value problem for the scalar linear advection equation with a constant characteristic speed, then define the geometric concept of characteristic curves and derive the exact solution. Next, we examine the solution to the Riemann problem as a special case. The effect of boundaries and the need for boundary conditions are then discussed, leading to the definition of the initial-boundary value problem, with relevant examples provided. The linear advection equation is then generalised to include source terms, variable characteristic speeds, and multiple spatial dimensions. We proceed by introducing general linear hyperbolic systems, providing definitions for eigenvalues, eigenvectors, and hyperbolicity. Characteristic variables are also introduced, and the general initial-value problem is solved analytically. The special case of the Riemann problem is revisited and solved exactly. A case study is included, consisting of a linear system of two equations derived from the linearisation of the nonlinear blood flow equations for arteries. We derive the eigenstructure and solve the general initial-value problem. Again, the Riemann problem is defined and solved, and a specific problem is solved in detail. Finally, a relaxation technique is introduced, whereby a parabolic advection-diffusion-reaction equation is approximated by a hyperbolic \(2 \times 2\) system of equations with stiff source terms. Exact solutions to a special initial-value problem are explicitly found. A carefully chosen set of exercises is provided at the end of the chapter.