The implementation of boundary conditions (BCs) can have a significant impact on the overall accuracy of a numerical simulation especially when dealing with compressible flows, either inviscid or viscous. It is therefore important to implement them correctly in CFD solvers. Usually BCs are imposed in two different ways: by directly modifying the boundary solution at each iteration/time-step (strong BCs) or by modifying the state from which the numerical flux is calculateda (weak BCs). Published studies to date suggest that weak BCs tend to improve convergence and can improve accuracy over strong BCs. 1, 7, 8 These studies are based upon steady simulations using predominantly loworder numerical methods, with successful application to high-order methods reported by Liu9 and Bassi. 3 In this paper we focus on a weak treatment of the BCs applied to the Euler and compressible Navier-Stokes equations within the context of compact discontinuous high-order spectral element methods (such as the discontinuous Galerkin (DG) method, and the high order Flux Reconstruction (FR) approach). In the Euler equations and the advection term of the compressible Navier-Stokes equations, the weak BCs are implemented by defining a left and a right state in order to solve a Riemann problem at the boundary and calculate the boundary intercell numerical flux via a Riemann solver, either exact or approximated. In Fig. 1, the boundary is represented such that the external (ghost) state is on the right, and the solution internal space is on the left. The ghost state is used as the right state of the Riemann solver at the boundary. The convention in Nektar++ is that the normal vector at a boundary, n, is outwardly directed from the interior of the solution domain and the component of the velocity perpendicular to the boundary is Vn= V· n. For Riemann invariant (characteristic), slip and no-slip conditions, we have adopted two different weak approaches. The first makes use of a Riemann solver, while the second directly calculate the intercell numerical flux at the given boundary by using the known solution. Hereafter we will refer to these two ways as Weak-Riemann and Weak-Prescribed approaches respectively.