Explaining the foundation of cognitive abilities in the processing of information by neural systems has been in the beginnings of biophysics since McCulloch and Pitts pioneered work within the biophysics school of Chicago in the 1940s and the interdisciplinary cybernetists meetings in the 1950s, inseparable from the birth of computing and artifcial intelligence. Since then, neural network models have traveled a long path, both in the biophysical and the computational disciplines. The biological, neurocomputational aspect reached its representational maturity with the Distributed Associative Memory models developed in the early 70 s. In this framework, the inclusion of signal-signal multiplication within neural network models was presented as a necessity to provide matrix associative memories with adaptive, context-sensitive associations, while greatly enhancing their computational capabilities. In this review, we show that several of the most successful neural network models use a form of multiplication of signals. We present several classical models that included such kind of multiplication and the computational reasons for the inclusion. We then turn to the diferent proposals about the possible biophysical implementation that underlies these computational capacities. We pinpoint the important ideas put forth by diferent theoretical models using a tensor product representation and show that these models endow memories with the context-dependent adaptive capabilities necessary to allow for evolutionary adaptation to changing and unpredictable environments. Finally, we show how the powerful abilities of contemporary computationally deep-learning models, inspired in neural networks, also depend on multiplications, and discuss some perspectives in view of the wide panorama unfolded. The computational relevance of multiplications calls for the development of new avenues of research that uncover the mechanisms our nervous system uses to achieve multiplication.