Distributed Computer Systems have become competitive in providing large amounts of computational power at a very low cost. The system consisting of personal computers, mini or main frames, or high performance multiprocessors, integrated into a high speed computer network is capable of providing any organization the power of a super machine with only a small initial cost. Such a system has the further benefit of providing industry with an easy and modular upgrade path because increasing the power of the system simply involves increasing the number of networked computers. Distributed Multimedia Applications (DMA) are so rich in their diversity of methodology and so inherently computational intensive that they naturally require a very heterogeneous mix of distributed processors interconnected by an efficient interconnection network. A DMA can be represented by a precedence graph, where nodes represent components (or modules) of the application, interconnected by arcs representing data streams flowing between different components. In this paper we study the problem of partitioning and mapping Distributed Multimedia Application graphs onto a heterogeneous distributed computer system. Such applications require a compromise between quality of service and cost of the utilized resources of the distributed computer system. As the problem is difficult to solve, in general, we present an approximate scheme which optimally assigns task modules of the application onto the processors of the distributed system.