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Gauging is a powerful operation on symmetries in quantum field theory (QFT),as it connects distinct theories and also reveals hidden structures in a giventheory. We initiate a systematic investigation of gauging discrete generalizedsymmetries in two-dimensional QFT. Such symmetries are described by topologicaldefect lines (TDLs) which obey fusion rules that are non-invertible in general.Despite this seemingly exotic feature, all well-known properties in gauginginvertible symmetries carry over to this general setting, which greatlyenhances both the scope and the power of gauging. This is established byformulating generalized gauging in terms of topological interfaces betweenQFTs, which explains the physical picture for the mathematical concept ofalgebra objects and associated module categories over fusion categories thatencapsulate the algebraic properties of generalized symmetries and theirgaugings. This perspective also provides simple physical derivations ofwell-known mathematical theorems in category theory from basic axiomaticproperties of QFT in the presence of such interfaces. We discuss abootstrap-type analysis to classify such topological interfaces and thus thepossible generalized gaugings and demonstrate the procedure in concreteexamples of fusion categories. Moreover we present a number of examples toillustrate generalized gauging and its properties in concrete conformal fieldtheories (CFTs). In particular, we identify the generalized orbifold groupoidthat captures the structure of fusion between topological interfaces(equivalently sequential gaugings) as well as a plethora of new self-dualitiesin CFTs under generalized gaugings.

Gauging Non-Invertible Symmetries: Topological Interfaces and Generalized Orbifold Groupoid in 2d QFT