In this chapter we derive the exact solution to the Riemann problem for the non-linear system of blood flow equations, under the assumptions that the Coriolis coefficient is \(\alpha =1\) , and with constant geometric and mechanical parameters. This solution serves as a reference for assessing numerical methods, applying boundary conditions, and designing Godunov-type numerical schemes. Additionally, it is valuable for devising and evaluating approximate Riemann solvers used in such numerical methods. Importantly, the exact solution to these non-linear problems enhances our understanding of the governing equations as a mathematical model for blood flow in compliant vessels. To proceed, one must also incorporate the tube law for the mechanics of the vessel wall, which distinguishes arteries from veins. First, we solve the Riemann problem for arteries, where the two blood flow equations are supplemented by a third equation for a passive scalar, representing the transport of chemical species. The main properties of the equations are reviewed, and rarefaction wave solutions are analysed using generalised Riemann invariants. An all-rarefaction exact solution is derived. Shock waves are introduced via the Rankine-Hugoniot conditions. The complete set of equations, encompassing rarefactions, shocks, and contacts, is presented, including a single non-linear algebraic equation for the cross-sectional area. It is shown that this equation has a unique solution within the physically valid range of positive cross-sectional areas. A numerical solution and a sampling procedure are implemented to obtain the full solution to the Riemann problem. Next, we solve the Riemann problem for veins. This problem is more complex than that for arteries, and all steps are rigorously justified. The overall methodology follows the same approach used for arteries, and sample solutions are provided. The chapter concludes with a summary and suggestions for further reading. A set of mini-projects is proposed for exercises based on the content of this chapter and related topics in the book.