In this chapter we address the question of how to stabilize high-order methods, focusing on the application of these methods to hyperbolic equations, particularly for wave propagation. All numerical methods suffer from aliasing errors, particularly methods that have little to no inherent dissipation such as high-order methods; we saw this in the dissipation-dispersion analysis of CG and DG methods in Ch. 7. In order to avoid numerical instabilities caused by such errors, high-order methods must use some form of stabilization mechanism, whereby this mechanism usually takes the form of dissipation. Note that this dissipation can be viewed in numerous ways, some not so favorably. However, another way of viewing these mechanisms is to acknowledge that energy must be allowed to cascade from the higher frequencies to the lower frequencies (Kolmogorov scales). This is in fact a very physical argument that confirms the need for such dissipation. For element-based Galerkin methods, the options available to us for introducing dissipation include: diffusion or hyper-diffusion (also called hyper-viscosity) operators that are tailored to only target high-frequency noise, low-pass filters which curtail the solution in the frequency domain, element-based limiters, and upwinding methods. In this chapter, we refer to diffusion-based stabilization as hyper-diffusion and only discuss upwinding methods for the discontinuous Galerkin method because it is essential to the general construction of DG methods since all DG methods must use a numerical flux and this flux, we have already seen in Chs. 6 and 7, performs best when it is based on upwinding methods. It should be noted that upwinding methods are also available for CG methods such as in the Petrov-Galerkin method [107] and in the relatively new discontinuous Petrov-Galerkin method [44]. Unlike the Bubnov-Galerkin method where the test and trial functions are always the same, the Petrov-Galerkin method uses different test and trial functions. We do not discuss Petrov-Galerkin methods in detail but cover other related upwinding-type methods for both CG and DG, such as the streamline upwinding Petrov-Galerkin (SUPG) method [48], the variational multi-scale (VMS) method [209], and a new stabilization known as the dynamic sub-grid scale (DSGS) method [268]. We also discuss a relatively new approach to stabilize numerical methods using an approach referred to as provably stable methods.