Construction of non-PT-symmetric complex potentials with all-real spectra

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.