In this chapter I review the work on the theory of associative memory that was carried out by myself in collaboration with O. P. Buneman and H. C. Longuet-Higgins at the Theoretical Psychology Unit, Edinburgh University, between 1967 and 1972. We were interested in the basic mathematical problems encountered in designing associative memory devices that would store their information in a nonlocal fashion. We were particularly interested in the design of memory models that could be implemented in neural tissue. Some of this work has already been published (Willshaw, 1972; Willshaw & Buneman, 1972; Willshaw, Buneman & Longuet-Higgins, 1969), but no overall review exists and many of the results in my thesis (Willshaw, 1971) have not been previously published.