  In most introductory courses on quantum mechanics one is taught that the
Hamiltonian operator must be Hermitian in order that the energy levels be real
and that the theory be unitary (probability conserving). To express the
Hermiticity of a Hamiltonian, one writes $H=H^\dagger$, where the symbol
$\dagger$ denotes the usual Dirac Hermitian conjugation; that is, transpose and
complex conjugate. In the past few years it has been recognized that the
requirement of Hermiticity, which is often stated as an axiom of quantum
mechanics, may be replaced by the less mathematical and more physical
requirement of space-time reflection symmetry (PT symmetry) without losing any
of the essential physical features of quantum mechanics. Theories defined by
non-Hermitian PT-symmetric Hamiltonians exhibit strange and unexpected
properties at the classical as well as at the quantum level. This paper
explains how the requirement of Hermiticity can be evaded and discusses the
properties of some non-Hermitian PT-symmetric quantum theories.
