In Chap.  2 , to approximate a function value at a mid-point or cell boundary with a specific order of accuracy, we assume that the function values at nodes or cell centers surrounding the mid-point (the function values on the stencil surrounding the mid-point) are all known. This assumption, while valid in the interior of a domain, is not necessarily valid for approximations near the domain’s boundaries. Here, we discuss approximations on and near boundaries in the context of numerical solutions of PDEs governing compressible fluid dynamics. To explain how a proper approximation near domain boundaries can be accomplished in a finite difference scheme, we first need to discuss the type of grid points we intend to use. Given a spatial domain, we may choose two types of grid points: (1) a set of grid points that overlap the domain’s boundary surfaces or (2) a set of grid points interior to the domain’s boundary surfaces. Let us choose the second type of grid where the grid points are interior to the domain’s boundary surfaces. This type of grid points facilitates the conservative approximation of discontinuous features such as shock wavesDiscontinuities and large gradientsshock waves near boundaries. In this chapter, we describe boundary treatments for Cartesian domainsCartesian domain (or those easily transformed to Cartesian ones) and general non-Cartesian domainsNon-Cartesian domain. We first discuss a general approach for imposing boundary conditions for equations such as Euler and Navier-Stokes equations [1, 2]. The approach is consistent with the dimensionality and physics of the underlying system, unlike the common one-dimensional, non-diffusive boundary condition enforcement [3]. This general characteristic approach will be used in boundary treatments for both Cartesian and non-Cartesian domainsNon-Cartesian domain.