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This talk will describe the synthesis and physics of semiconductor nanomaterials--inorganic ribbons/membranes and single walled carbon nanotubes-that are structured into wavy geometries through engineering control over nonlinear buckling instabilities.Experimental measurements on various systems of this general type, together with analytical and finite element modeling of their responses, reveal the essential physics.Use of these wavy semiconductors in high performance field effect transistors, pn junction diodes, and even fully formed integrated circuits on elastomeric supports illustrates pathways to electronic systems that offer full stretchability, with purely elastic responses to applied strains of up to ~100%.Hemispherical electronic eye imagers, conformable sensor skins and other devices that rely on this type of approach will be discussed.