A major paradigm of 20th-century science is to understand nature in the language of quantum field theory. Efforts to answer foundational questions about this language have led to successful and ongoing cross-fertilisation between theoretical physics and pure mathematics. In particular, Atiyah and Segal proposed an axiomisation of the path integral by beautifully linking geometry with algebra. The talk starts with a lightening review of this functorial approach, and then quickly restricts to the case in which spacetime is two-dimensional and has no geometric structure: two-dimensional topological quantum field theory (TQFT). This seemingly simple situation is still surprisingly rich, and we will see how algebras, categories, and "higher" structures appear naturally; examples of such structures are ubiquitous in theoretical physics, string theory, and many areas of mathematics. Once the stage is carefully set, we turn to the central notion of symmetry, which involves the action of groups on a TQFT. We will be led to interpret symmetries as special kinds of "defects" of the TQFT, which in turn allows for a natural, purely algebraic generalisation of orbifolding. This leads to new equivalences between TQFTs, of which we will discuss the examples of Landau-Ginzburg models, and (if time permits) refined knot invariants.