Abstract: What is the most powerful topological quantum field theory (TQFT)? And, which 4-manifold invariants can detect the Gluck twist? Guided by questions like these, we will look for new invariants of 3-manifolds and smooth 4-manifolds. Traditionally, a construction of many such invariants and TQFTs involves a choice of certain algebraic structure, so that one can talk about “invariants for SU(2)” or a “TQFT defined by a given Frobenius algebra.” Surprisingly, recent developments lead to an opposite phenomenon, where algebraic structures are labeled by 3-manifolds and 4-manifolds, so that one can speak of VOA-valued invariants of 4-manifolds or MTC-valued invariants of 3-manifolds. Explaining these intriguing connections between topology and algebra will be the main goal of these lectures.