  Gauging is a powerful operation on symmetries in quantum field theory (QFT),
as it connects distinct theories and also reveals hidden structures in a given
theory. We initiate a systematic investigation of gauging discrete generalized
symmetries in two-dimensional QFT. Such symmetries are described by topological
defect lines (TDLs) which obey fusion rules that are non-invertible in general.
Despite this seemingly exotic feature, all well-known properties in gauging
invertible symmetries carry over to this general setting, which greatly
enhances both the scope and the power of gauging. This is established by
formulating generalized gauging in terms of topological interfaces between
QFTs, which explains the physical picture for the mathematical concept of
algebra objects and associated module categories over fusion categories that
encapsulate the algebraic properties of generalized symmetries and their
gaugings. This perspective also provides simple physical derivations of
well-known mathematical theorems in category theory from basic axiomatic
properties of QFT in the presence of such interfaces. We discuss a
bootstrap-type analysis to classify such topological interfaces and thus the
possible generalized gaugings and demonstrate the procedure in concrete
examples of fusion categories. Moreover we present a number of examples to
illustrate generalized gauging and its properties in concrete conformal field
theories (CFTs). In particular, we identify the generalized orbifold groupoid
that captures the structure of fusion between topological interfaces
(equivalently sequential gaugings) as well as a plethora of new self-dualities
in CFTs under generalized gaugings.
