Orbifold data are higher algebras in topological quantum field theories which can be gauged to obtain a new TQFT. Examples include gaugings of (higher) group actions, state sum models, and more generally gaugings of "non-invertible symmetries". The defining conditions of orbifold data algebraically encode invariance under choices of triangulations used in the gauging process, which has been rigorously developed in arbitrary dimensions. The talk gives an introduction to TQFT and the orbifold construction, and it provides examples in dimensions 2,3, and 4.