This chapter presents high-order, non-linear numerical methods for solving non-linear systems of hyperbolic equations with source terms, written in both conservative and non-conservative forms. Two major classes of methods are discussed: fully-discrete and semi-discrete methods. From the fully-discrete class, we study ADER schemes, which are based on non-linear ENO/WENO spatial reconstructions using polynomials of degree m. Their space-time coupling is achieved through solving the generalised Riemann problem \(GRP_{m}\) for the numerical flux and a similar procedure for the integral of the source term. The resulting schemes achieve accuracy of \(m+1\) in both space and time, where m is arbitrary. Semi-discrete methods discretise the PDEs separately in space and time. These methods use ENO/WENO spatial reconstructions and, typically, TVD Runge-Kutta time integrators for the associated ordinary differential equations in time. As examples, we provide detailed descriptions of various fully-discrete and semi-discrete methods of second-order accuracy for both conservative and non-conservative systems. We also examine fully-discrete ADER methods of arbitrary accuracy in both space and time for both conservative and non-conservative systems. Additionally, semi-discrete schemes of second- and third-order accuracy are presented for conservative systems of balance laws. Computational results for selected test problems are provided for four of the methods studied. First, a convergence rate study is conducted to verify that the theoretically expected orders of accuracy are attained. Results for schemes of second to fifth order are presented. The same schemes are also evaluated through Riemann problems with exact solutions, involving rarefactions, shocks, and contact discontinuities. The chapter concludes with a summary, conclusions, and suggestions for further reading.