A new process for storing strings of objects on a best match basis is defined. This process is used to construct a simple computer system and more complex systems compounded from this. These systems are compared with some models of psychological processes. It is shown that some elementary programming constructs can be duplicated by these systems, that others are impossible, and that even the simplest system can emulate any given Turing machine. Particular consideration is given to the fact that the memory of the system is continually changing and being updated. In an analysis which has not been undertaken elsewhere it is shown that a fixed algorithm can be encoded even within this changing memory. This result is particularly important as it strongly constrains the possible structures for the multiple systems. General principles underlying both natural and artificial adaptive systems are indicated by these results. A number of ways of implementing the system on both serial and parallel digital computers are compared in terms of storage efficiency and execution times. An actual implementation is exhibited and used to support these theoretical estimates. It is concluded that execution times almost independent of memory size can be obtained in practice. An alternative best matching algorithm which finds nearest neighbours in Euclidean space is examined. A new analysis of this algorithm is given which shows that its execution time is independent of the number of points being matched but that it increases exponentially with the dimension of the Euclidean space. A new algorithm is presented for solving some systems of simultaneous equations over finite domains. This algorithm can be used for extracting additional information from a best match memory. However, it is cast in a very general form and should be applicable elsewhere.