This talk focuses on recent developments on the definition of boundary conditions for simulations of compressible flow. Boundary conditions which are physically realistic and mathematically well-posed are of critical importance to the simulation of many compressible-flow phenomena. In particular, it is often desirable to compute a flow over only a small region of interest, under the assumption that flow in adjacent regions will not have a significant impact on the solution. Mathematically and operationally, this statement needs to be made precise, and, in the process, we are often led to the choice of nonreflecting (or absorbing) boundary conditions. Nonreflecting boundary conditions are often based on a linearization about a uniform flow. Recently, we have developed local, well-posed boundary conditions for the linearized Euler equations (indeed any hyperbolic equations) which can be extended to arbitrarily high-order of nonreflectivity. An important aspect of these boundary conditions is that they explicitly account for the dispersive nature of finite difference approximations. Thus we are able to avoid reflection of not only well-resolved waves, but also of "spurious" waves which are an artifact of the discretization. Such spurious waves have been shown to lead to inappropriate self-forcing of convectively-unstable flows. Finally, in nonlinear computations where large (turbulent) fluctuations must cross a computational boundary, the accuracy of nonreflecting boundary conditions is degraded. Various fixes for this situation will be discussed. Several example of the use of artificial boundary conditions will be given. These include computations of sound generation by turbulence in mixing layers and jets, computation of self-sustained oscillations in globally unstable flows, such as open cavities, and computations of bubbly cavitating flows.