  Topological phases in (2+1)-dimensions are frequently equipped with global
symmetries, like conjugation, bilayer or electric-magnetic duality, that
relabel anyons without affecting the topological structures. Twist defects are
static point-like objects that permute the labels of orbiting anyons. Gauging
these symmetries by quantizing defects into dynamical excitations leads to a
wide class of more exotic topological phases referred as twist liquids, which
are generically non-Abelian. We formulate a general gauging framework,
characterize the anyon structure of twist liquids and provide solvable lattice
models that capture the gauging phase transitions. We explicitly demonstrate
the gauging of the $\mathbb{Z}_2$-symmetric toric code, $SO(2N)_1$ and
$SU(3)_1$ state as well as the $S_3$-symmetric $SO(8)_1$ state and a
non-Abelian chiral state we call the "4-Potts" state.
