Abstract: In recent years, the assumption that quantum mechanical operators be represented by Hermitian matrices has been examined, and found to be sufficient but not necessary in order to construct a viable theory of quantum mechanics. Most notably, progress has occurred within the `PT quantum mechanics' framework developed by Bender et al. In this regime, all the virtues of Hermiticity are achieved by means of the parity (P) and time-reversal (T) operators. In this talk I will briefly review the formalism of PT quantum mechanics and present several applications of general interest. For example we find that the non-Hermitian generalization of the Dirac equation possesses several interesting features, including a toy model of massless (Dirac) neutrinos which nonetheless permit flavor oscillation. Another application arises in the realm of theoretical condensed matter physics; following a 1956 paper by Freeman Dyson, where he considered a non-Hermitian Hamiltonian in the context of ferromagnetism, we are motivated to consider the phenomenon of high temperature superconductivity, one of the most outstanding problems in theoretical physics.