Puckering in planar molecules and graphene-like 2D and quasi 2D systems. Special attention has been paid recently to 2D systems in view of a variety of their planar-surface-specific physical and chemical properties and possible graphene-like applications in electronics. Similar-to- graphene properties are sought for in silicene, phosphorene, boron nitride, zinc oxide, gallium nitride, as well as in 2D transition metal dichalkogenides and oxides, plus a number of other organic and inorganic 2D and quasi-2D compounds with expected similar properties. One of the main important features of these systems is their planarity or quasi-planarity, but many of the quasi-2D compounds are subject to out-of-plane deviations known as puckering (buckling). The instability and distortions of the planar configuration (as in any other systems in nondegenerate state) was shown to be due to the PJTE. Detailed exploration of the PJTE in such systems allows one to identify the excited states that are responsible for the puckering, and suggest possible external influence that restores their planarity, including oxidation, reduction, substitutions, or coordination to other species. Recent investigations have also extended to 3D compounds.