This chapter focuses on high-order numerical methods for scalar equations in both the finite volume and discontinuous Galerkin FE frameworks. We present two classes of methods for approximating hyperbolic equations with high-order accuracy: fully-discrete methods and semi-discrete methods. In the finite volume setting both classes rely on the non-linear spatial reconstruction methodology, in view of Godunov’s theorem.  ADER method Fully-discrete methods are represented here by the ADER methodology. These methods, within the finite volume framework, require a non-linear reconstruction procedure using polynomials of degree m, along with solving the  Generalised Riemann problem generalised Riemann problem \(GRP_{m}\) to compute a high-order numerical flux. The numerical source is calculated from a volume integral, evaluated on the time-evolved reconstruction polynomial within the volume. Semi-discrete methods is represented here by ENO/WENO spatial reconstructions combined with TVD Runge-Kutta schemes for solving the associated ordinary differential equations. The performance of the numerical methods is thoroughly assessed through a carefully selected suite of test problems, which include isolated smooth profiles, isolated discontinuous profiles, and mixed smooth/discontinuous multi-wave profiles. Results are presented for schemes with accuracy up to the 10th order in both space and time. The assessment includes convergence rate studies for smooth profiles, resolution of discontinuous profiles, and efficiency (i.e., error versus cost). Two key issues are worth noting. First, the higher-order range (from 5th to 10th order) reveals limitations of existing spatial reconstruction schemes in handling discontinuous solutions. Second, higher-order methods are orders of magnitude more efficient than low-order methods. Consequently, high-order methods are essential in the active field of modelling and simulation in science and engineering. A list of suggested exercises is provided at the end of the chapter.