Summary: The condition of self-adjointness ensures that the eigenvalues of a Hamiltonian are real and bounded below. Replacing this condition by the weaker condition \(\mathcal P\mathcal T\) of symmetry, one obtains new infinite classes of complex Hamiltonians whose spectra are also real and positive. These \(\mathcal P\mathcal T\) 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.