  We describe recent progress in understanding the continuous symmetry
properties of non-Hermitian, PT-symmetric quantum field theories. Focussing on
a simple non-Hermitian theory composed of one complex scalar and one complex
pseudoscalar, we revisit the derivation of Noether's theorem to show that the
conserved currents of non-Hermitian theories correspond to transformations that
do not leave the Lagrangian invariant. We illustrate the impact that this has
on the consistent formulation of (Abelian) gauge theories by studying a
non-Hermitian extension of scalar quantum electrodynamics. We consider the
spontaneous breakdown of both global and local symmetries, and describe how the
Goldstone theorem and the Englert-Brout-Higgs mechanism are borne out for
non-Hermitian, PT-symmetric theories.
