This paper addresses the problem of task assignment in distributed systems. A distributed system is defined as any configuration of two or more processors, each with private memory, in which computations utilise the combined resources of the component machines. A distributed process is defined as a set of tasks which together work towards some common goal. Each task spends a portion of its time excuting on one of the processors and a portion of its time communicating with other tasks to processors designates one processor for each task to reside on for its lifetime. We consider three different performance goals (cost functions) and investigate the problem of achieving optimalassignments with respect to each of them functions. In particular, we investigate task assignment to minimise (1) total execution and comminication costs, (2) total execution, communication and interference costs and (3) total execution and communication costs with bounds on the number of tasks assignedto each processor. In all cases the problem of finding an optimal assignmentfor an arbitrary number of processors is found to be NP-complete. This thesisfocuses on the development and simulation of suboptimal algorithms and on consideration of special cases for each of the three performance criteria.