Computational approaches are being developed to study a range of problems in aeroacoustics. These aeroacoustic problems may be classified based on the physical processes responsible for the sound radiation, and range from linear problems of radiation, refraction, and scattering in known base flows or by solid bodies, to sound generation by turbulence. In this article, we focus mainly on the challenges and successes associated with numerically simulating sound generation by turbulent flows. We discuss a hierarchy of computational approaches that range from semi-empirical schemes that estimate the noise sources using mean-flow and turbulence statistics, to high-fidelity unsteady flow simulations that resolve the sound generation process by direct application of the fundamental conservation principles. We stress that high-fidelity methods such as Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES) have their merits in helping to unravel the flow physics and the mechanisms of sound generation. They also provide rich databases for modeling activities that will ultimately be needed to improve existing predictive capabilities. Spatial and temporal discretization schemes that are well-suited for aeroacoustic calculations are analyzed, including the effects of artificial dispersion and dissipation on uniform and nonuniform grids. We stress the importance of the resolving power of the discretization as well as computational efficiency of the overall scheme. Boundary conditions to treat the flow of disturbances in and out of the computational domain, as well as methods to mimic anechoic domain extension are discussed. Test cases on some benchmark problems are included to provide a realistic assessment of several boundary condition treatments. Finally, highlights of recent progress are given using selected model problems. These include subsonic cavity noise and jet noise. In the end, the current challenges in aeroacoustic modeling and in simulation algorithms are revisited with a look toward the future developments. r 2004 Elsevier Ltd. All rights reserved. e front matter r 2004 Elsevier Ltd. All rights reserved. erosci.2004.09.001 ing author. Tel.: +1626 395 4021; fax: +1 626 568 2719. ess: firstname.lastname@example.org (T. Colonius).