Topological phases in two dimensions support anyonic quasiparticle excitations that obey neither bosonic nor fermionic statistics. These anyon structures often carry global symmetries, such as conjugation, bilayer or electric-magnetic duality, that relate distinct anyons with similar fusion and statistical properties. Anyonic symmetries associate topological defects or fluxes in topological phases. These twist defects are point-like objects that permute the anyon types orbiting quasiparticles according to the symmetry. As the symmetries are global and static, these extrinsic defects are semiclassical objects that behave disparately from conventional quantum anyons. Remarkably, even when the topological states supporting them are Abelian, they are generically non-Abelian and powerful enough for topological quantum computation. Gauging the global symmetries by quantizing twist defects into dynamical excitations leads to a wide class of more exotic topological phases referred as twist liquids, which are generically non-Abelian. In this lecture series, I review the theory of global symmetries in topological phases, and the general framework describing twist defects and twist liquids.