Explore the fascinating world of topologically ordered matter in this comprehensive colloquium talk by Steven H. Simon from Oxford University. Delve into the intriguing concept of two-dimensional systems where processes are governed by gross topology rather than detailed geometry. Discover how particle world lines in 2+1 dimensions can be interpreted as knots or links, leading to topological invariants in certain process amplitudes. Learn about the practical implications of this seemingly exotic physics, including its observed presence in quantum Hall experiments and potential applications in quantum computing. Examine proposed possibilities for creating similar physics in various systems, from superfluid helium and topological superconductors to semiconductor-superconductor junctions, quantum wires, spin systems, graphene, and cold atoms.