This chapter focuses on systems of non-linear hyperbolic equations expressed in non-conservative form. In several areas of scientific and technological interest, mathematical models cannot be represented in conservation-law form, even if they are derived from physical conservation principles. For such systems, the established class of conservative methods does not apply, and alternative approaches are required. The chapter begins with a motivating example, using a non-linear scalar equation, to highlight some of the issues that arise in the presence of discontinuities. The chapter introduces theoretical concepts related to general hyperbolic systems in non-conservative form, including non-conservative products, path functions, and generalised Rankine-Hugoniot conditions. A relevant example is provided by the blood flow equations, incorporating tube laws for arteries and veins, as well as discontinuous mechanical and geometric parameters. A framework for constructing numerical methods for non-conservative systems is introduced, specifically the path-conservative approach. In this methodology, the numerical fluctuation replaces the role of the numerical flux in the conservative case. Several non-conservative methods are examined, including upwind and centred schemes. These methods are applicable to the blood flow equations in arteries and veins, accommodating both constant and discontinuous geometric and biomechanical parameters. The key challenge in these methods is the construction of the numerical fluctuation. A systematic evaluation of the performance of these schemes is conducted using a carefully selected suite of test problems with exact solutions. Numerical results are presented for four of the seven methods studied in this chapter. High-order extensions of these methods are discussed in subsequent chapters. The chapter concludes with a summary, recommendations for further reading, and a list of suggested exercises.