  It is generally assumed that a Hamiltonian for a physically acceptable
quantum system (one that has a positive-definite spectrum and obeys the
requirement of unitarity) must be Hermitian. However, a PT-symmetric
Hamiltonian can also define a physically acceptable quantum-mechanical system
even if the Hamiltonian is not Hermitian. The study of PT-symmetric quantum
systems is a young and extremely active research area in both theoretical and
experimental physics. The purpose of this Review is to provide established
scientists as well as graduate students with a compact, easy-to-read
introduction to this field that will enable them to understand more advanced
publications and to begin their own theoretical or experimental research
activity. The ideas and techniques of PT symmetry have been applied in the
context of many different branches of physics. This Review introduces the
concepts of PT symmetry by focusing on elementary one-dimensional PT-symmetric
quantum and classical mechanics and relies in particular on oscillator models
to illustrate and explain the basic properties of PT-symmetric quantum theory.
