In the past few years, together with several collaborators, we have developed a framework linking PAC learning theory to topological combinatorics. At its core lies the notion of the spherical dimension of a concept class. Since the most established complexity measure in PAC learning is VC dimension, a natural question arises: can spherical dimension be bounded in terms of VC dimension? This question is compelling from both the learning-theoretic and the topological standpoint. In this talk, I will outline how the connection between these two dimensions arises, and survey what we know, and don't know, about their relation.