Global symmetries are fundamental to our understanding of nature: they organize the spectrum of physical theories, constrain their dynamics, and reveal emergent effective degrees of freedom at low energies. Crucially, global symmetries are non-perturbative tools that shed light on strongly coupled regimes inaccessible to perturbative methods. Recently, it was realized that symmetries have an intrinsic formulation in quantum field theory through topological operators, also known as symmetry defects. This perspective has led to important generalizations, including the discovery of non-invertible symmetries, characterized by categorical fusion rules rather than conventional group multiplication. In this thesis, we investigate these non-invertible symmetries in the context of finite-group gauge theories, which are exactly solvable models and serve as prototypical examples of topological quantum field theories. %We show how the folding trick can be used to classify certain non-invertible symmetry defects in these theories, compute their fusion rules, and determine their action on other operators. Finally, we identify a non-invertible generalization of electric–magnetic duality, extending this notion from abelian to non-abelian gauge groups. In the first part of this work (Chapter 2) we investigate the invertible and non-invertible symmetries of topological finite-group gauge theories in general spacetime dimensions, where the gauge group can be abelian or non-abelian. We focus in particular on the 0-form symmetry. The gapped domain walls that generate these symmetries are specified by boundary conditions for the gauge fields on either side of the wall. We investigate the fusion rules of these symmetries and their action on other topological defects, including the Wilson lines, magnetic fluxes, and gapped boundaries. We illustrate these constructions with various novel examples, including non-invertible electric-magnetic duality symmetry in 3+1d $\mathbb{Z}_2$ gauge theory and non-invertible analogs of electric-magnetic duality symmetry in non-abelian finite-group gauge theories. In particular, we discover topological domain walls that obey Fibonacci fusion rules in 2+1d gauge theory with dihedral gauge group of order 8. We also generalize the Cheshire string defect to analogous defects of general codimensions and gauge groups and show that they form a closed fusion algebra. In the second part (Chapter 3) we demonstrate how to realize these symmetries as condensation defects, i.e., as suitable insertions of lower dimensional topological operators. We then compute these symmetries' fusion rules and action using their condensation expression and the algebraic properties of the lower-dimensional objects that make them. We illustrate the discussion in $\mathbb{Z}_N$ gauge theory, where we derive the correspondence between domain walls, labeled by subgroups and topological actions for the doubled gauge group, and higher gauging condensation defects, labeled by subalgebras of the global symmetry. As a primary application, we obtain the condensation expression for the invertible symmetries of abelian gauge theories defined by outer automorphisms of the gauge group. We also show how to use these ideas to derive the action for certain non-abelian groups. For instance, one can obtain the action for the Dihedral group $\mathbb{D}_4$ by gauging a swap symmetry of $\mathbb{Z}_2\times\mathbb{Z}_2$ gauge theory.