A gauge theory has internal degrees of freedom that do not affect the foretold physical outcomes of the theory. The theory has a Lie group of *continuous symmetries* of these internal degrees of freedom, *i.e.* the predicted physics under any transformation in this group on the degrees of freedom. Examples include the $U(1)$-symmetric quantum electrodynamics and other Yang-Mills theories wherein non-Abelian groups replace the $U(1)$ gauge group of QED.

Gauge theories are divided into two categories. A local gauge theory has internal degrees of freedom that do not affect the foretold physical outcomes of the theory. The theory has a Lie group of continuous symmetries of these internal degrees of freedom, i.e. the predicted physics under any transformation in this group on the degrees of freedom. Examples include the $U(1)$-symmetric quantum electrodynamics and other Yang-Mills theories wherein non-Abelian groups replace the $U(1)$ gauge group of QED.

There are also gauge theories with a globally non-trivial structure that allow for physical dependence on the "gauge". The Aharanov Bohm and Berry Phase are famous examples of such Large Gauge Theories.

The gauge groups of such theories are proposed to model experimentally observed symmetries of the described physical processes. These symmetries can include observed conserved quantities in experiments. The requirement to have a certain gauge group to explain observed symmetries guides the theorist towards possible theoretical descriptions: e.g. to a Lagrangian invariant with respect to transformations belonging to the gauge group. The gauge group of the Standard Model is $SU(3)\times SU(2) \times U(1)$, where $SU(3)$ is the gauge group of quantum chromodynamics and $SU(2) \times U(1)$ the gauge group of the electroweak force.

Mathematically, a gauge theory is a fiber bundle with the observable physical model as the bundle's base space and the structure group as the group of gauge symmetries. A lot of times these fiber bundles can be trivializable, allowing for a global section that enables a local description, but over 4 manifolds, even simply connected Lie Groups can be shown to always have a non-trivial description via obstruction theory. In a trivial fiber bundle, the fibers represent the "unphysical" (unobservable) degrees of freedom.