To a large extent, the pursuit of science takes place in universities. In this essay, I ask the following questions. Supposing there were no universities, and that all the knowledge mankind has ever collected and generated is somehow accessible, would we invent universities to make this knowledge available to address the problems humanity faces? What should those universities perform, and what role would science play in such universities? To look for answers to those questions, I consider the nature of the problems dealt with by science, the knowledge needed to address those problems, the gap between the two, the need for interdisciplinarity and the need to educate the leaders of the future, and finally, the boundaries of scientific knowledge.