In physics, topological quantum field theories (TQFT) are used to describe quantum mechanical systems that depend only on global, topological aspects of spacetime. At the same time, they provide a crucial tool for characterizing exotic, non-group-theoretic symmetries of higher-dimensional systems.

In pure mathematics, topological quantum field theories lead to an impressive unification of manifold topology and algebra: every TQFT results in a manifold invariant, and conversely, TQFTs allow the transfer of methods and results from the study of low-dimensional manifolds to purely algebraic problems.

After briefly introducing the definition and basic examples, in this talk I will highlight the underlying algebraic structures, and conversely the sensitivity of such TQFTs to manifold topology. I also discuss applications in physics and computer science.