  We review recent work on the generalization of PT symmetry. We show that, in
addition to PT-symmetric complex potentials, there are also large classes of
non-PT-symmetric complex potentials which also feature all-real spectra. In
addition, some classes of these non-PT-symmetric potentials allow phase
transitions which do or do not go through exceptional points. These
non-PT-symmetric potentials are constructed by a variety of methods, such as
the symmetry and supersymmetry methods and the soliton theory. A generalization
of PT symmetry in multi-dimensions is also reviewed.
