  I will provide a pedagogical introduction to non-Hermitian quantum systems
that are PT-symmetric, that is they are left invariant under a simultaneous
parity transformation (P) and time-reversal (T). I will explain how generalised
versions of this antilinear symmetry can be utilised to explain that these type
of systems possess real eigenvalue spectra in parts of their parameter spaces
and how to set up a consistent quantum mechanical framework for them that
enables a unitary time-evolution. In the second part I will explain how to
extend this framework to explicitly time-dependent Hamiltonian systems and
report in particular on recent progress made in this context. I will explain
how to construct the essential key quantity in this framework, the
time-dependent Dyson map and metric and solutions to the time-dependent
Schr\"odinger equation, in an algebraic fashion, using time-dependent Darboux
transformations, utilising Lewis-Riesenfeld invariants, point transformations
and some approximation methods. I comment on the ambiguities of this metric and
demonstrate that this can even lead to infinite series of metric operators. I
conclude with some applications to PT-symmetrically coupled oscillators,
demonstrate the equivalence of the time-dependent double wells and unstable
anharmonic oscillators and show how the unphysical PT$-symmetrically broken
regions in the parameter space for the time-independent theory becomes physical
in the explicitly time-dependent systems. I discuss how this leads to a
prolongation of the otherwise rapidly decaying von Neumann entropy. The
so-called sudden death of the entropy is stopped at a finite value.
