Explore quantum error-correcting codes and their fundamental limits in this lecture from Microsoft Research's Jeongwan Haah at the Park City Mathematics Institute. Delve into the Quantum Singleton bound and its implications for practical quantum computing, with particular focus on the surface code - a critical quantum error correction method relevant to both practical applications and topological phases of matter. Learn why local codes in one dimension face limitations in achieving large code distance, understand how two-dimensional local codes relate to linear system size, and discover the constraints on logical qubits in d-dimensional local codes. Examine why two-dimensional translation invariant codes invariably become surface codes, and gain insights into triorthogonal codes used in magic state distillation. Part of a comprehensive graduate summer school program featuring leading experts in quantum computation, this lecture includes accompanying problem sessions designed to reinforce understanding of these advanced quantum computing concepts.