  Topological phases in two dimensions support anyonic quasiparticle
excitations that obey neither bosonic nor fermionic statistics. These anyon
structures often carry global symmetries that relate distinct anyons with
similar fusion and statistical properties. Anyonic symmetries associate
topological defects or fluxes in topological phases. 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. In this
article, I review the most recent theoretical developments on symmetries and
defects in topological phases.
