Periodic systems of molecules are charts of molecules similar to the periodic table of the elements. Construction of such charts was initiated in the early 20th century and is still ongoing. It is commonly believed that the periodic law, represented by the periodic chart, is echoed in the behavior of molecules, at least small molecules. For instance, if one replaces any one of the atoms in a triatomic molecule with a rare gas atom, there will be a drastic change in the molecule’s properties. Several goals could be accomplished by constructing an explicit representation of this periodic law as manifested in molecules: (1) a classification scheme for the vast number of molecules that exist, starting with small ones having just a few atoms, for use as a teaching aid and tool for archiving data, (2) forecasting data for molecular properties based on the classification scheme, and (3) a sort of unity with the periodic chart and the periodic system of fundamental particles. Periodic systems (or charts or tables) of molecules are the subjects of two reviews. The systems of diatomic molecules include those of (1) H. D. W. Clark, and (2) F.-A. Kong, which somewhat resemble the atomic chart. The system of R. Hefferlin et al. was developed from (3) a three-dimensional to (4) a four-dimensional system Kronecker product of the element chart with itself. A totally different kind of periodic system is (5) that of G. V. Zhuvikin, which is based on group dynamics. In all but the first of these cases, other researchers provided invaluable contributions and some of them are co-authors. The architectures of these systems have been adjusted by Kong and Hefferlin to include ionized species, and expanded by Kong, Hefferlin, and Zhuvikin and Hefferlin to the space of triatomic molecules. These architectures are mathematically related to the chart of the elements. They were first called “physical” periodic systems.