Given a collection of convex open sets in Euclidean space, one can form an associated combinatorial code (a subset of the Boolean lattice). Which codes arise this way? This question has been a rich and active area of research since it was first posed by Curto et al in 2012. I will speak about a recent approach to this question which involves defining and investigating certain "minimally non-convex" codes. Developing this framework has led to several new results and techniques of independent interest, as well as natural new questions. I will describe the story so far, as well as future directions of exploration. Along the way we will see numerous connections to existing areas of research such as representability of simplicial complexes, Helly-type theorems, lattice theory, and other combinatorial ideas.