The study of fundamental groups of compact positively curved spaces is relevant to two general questions: Which smooth manifolds admit metrics with positive curvature, and which groups appear as fundamental groups of such spaces? Chern (suggesting an analogue of Preissman's theorem in the negatively curved case) conjectured that the only abelian subgroups are cyclic, though this was later shown to not be the case. In this talk, I will give an overview of what is known, drawing on (by now) classical work of Synge and Grove-Searle and recent work of Wilking, Rong, and others who have proven the cyclicity of (or the cyclicity of abelian subgroups of) fundamental groups of positively curved spaces with high rank isometry group.