The search for non-Abelian phases of matter has inspired decades of groundbreaking research in fractional quantum Hall systems. The strongest experimental evidence of such phases arises at half-integer filling factors. I will briefly introduce the landscape of competing states in these systems and the key obstacles to identifying their topological order. I will then describe recent theoretical and experimental advances, which include novel routes for resolving these enigmatic states and reveal a systematic pattern of non-Abelian phases in bilayer graphene.