  The condition of self-adjointness ensures that the eigenvalues of a
Hamiltonian are real and bounded below. Replacing this condition by the weaker
condition of ${\cal PT}$ symmetry, one obtains new infinite classes of complex
Hamiltonians whose spectra are also real and positive. These ${\cal PT}$
symmetric theories may be viewed as analytic continuations of conventional
theories from real to complex phase space. This paper describes the unusual
classical and quantum properties of these theories.
