Mechanical buckling usually means catastrophic failure in structural mechanics systems. However, controlled buckling of thin films on compliant substrates has been used to advantage in diverse fields such as micro-/nanofabrication, optics, bioengineering, and metrology as well as fundamental mechanics studies. In this Feature Article, a mechanical buckling model is presented, which sprang, in part, from the buckling study of high-quality, single-crystalline nanomaterials. To check the mechanical-buckling phenomenon down to the nano-/molecular scale, well-aligned single-walled carbon nanotube arrays and cross linked carbon-based monolayers are transferred from growth substrate onto elastomeric substrate and then they are buckled into well-defined shapes that are amenable to quantitative analysis. From this nano- or molecular-scale buckling, it is shown that the mechanical moduli of nanoscale materials can easily be determined, even using a model based on continuum mechanics. in addition, buckling phenomena can be utilized for the determination of mechanical moduli of organic functional materials such as poly(3-hexylthiophene) (P3HT) and P3HT/6,6-phenyl-C-61-butyric acid methyl ester (PCBM) composite, which are widely used for organic transistors and organic photovoltaics. The results provide useful information for the realization of flexible and/or stretchable organic electronics. Finally, the fabrication and applications of "wavy, stretchable" single-crystal Si electronics on elastomeric substrates are demonstrated.