Synthesis using physical modeling has a long history. As computational costs for physical modeling synthesis are often much greater than for conventional synthesis methods, most techniques currently rely on simplifying assumptions. These include digital waveguides, as well as modal synthesis methods. Although such methods are efficient, it can be difficult to approach some of the more detailed behavior of musical instruments in this way, including strongly nonlinear interactions. Mainstream time-stepping simulation methods, despite being computationally costly, allow for such detailed modeling. In this article, the results of a five-year research project, Next Generation Sound Synthesis, are presented, with regard to algorithm design for a variety of sound-producing systems, including brass and bowed-string instruments, guitars, and large-scale environments for physical modeling synthesis. In addition, 3-D wave-based modeling of large acoustic spaces is discussed, as well as the embedding of percussion instruments within such spaces for full spatialization. This article concludes with a discussion of some of the basics of such time-stepping methods, as well as their application in audio synthesis.