Inferring the evolutionary history of cognitive abilities requires large and diverse samples. However, such samples are often beyond the reach of individual researchers or institutions, and studies are often limited to small numbers of species. Consequently, methodological and site-specific-differences across studies can make comparisons between species problematic. Here we introduce the ManyPrimates project, which addresses these challenges by providing a large-scale collaborative framework for comparative studies in primate cognition. To demonstrate the viability of the project we conducted a case study of short-term memory. In this initial study, we were able to include 176 individuals from 12 primate species housed at 11 sites across Africa, Asia, North America and Europe. All subjects were tested in a delayed-response task using consistent methodology across sites. Individuals could access food rewards by remembering the position of the hidden reward after a 0, 15, or 30-second delay. Overall, individuals performed better with shorter delays, as was predicted by previous studies. The phylogenetic analysis we conducted showed a strong phylogenetic signal for short-term memory abilities. However, with only 12 species, the validity of this analysis is limited. Our initial results demonstrate the feasibility of a large, collaborative open-science project, and we present the ManyPrimates project as an exciting opportunity to address open questions in primate cognition and behaviour with large, diverse datasets.