The lattice-mismatch-induced strain in growth of Ge on Si produces a host of exciting scientific and technological consequences, both in 3D nanostructure formation and, when silicon-on-insulator (SOI) is used as a substrate, in 2D membrane fabrication. One can use the ideas of strain sharing and critical thickness, combined with the ability to release the top layers of SOI, to create free-standing, dislocation-free, elastically strain relieved flexible Si/Ge membranes with nanometer scale thickness, which we call NanoFLEXSi or Si nanomembranes (SiNMs). The membranes can be transferred to new substrates, producing the potential for novel heterogeneous integration. The very interesting, and in some cases surprising, structural and electronic properties of these very thin membranes have been revealed using STM, x-ray diffraction, and electronic transport measurements. For example, STM shows that conduction in very thin Si layers on SOI with bulk-Si mobilities is possible even though the membrane is bulk depleted. Using the effect of elastic strain, we have fabricated two-dimensional electron gases (2DEGs) in membrane structures; we support the transport measurements with calculations suggesting that we are observing a single bound state in the well. We have fabricated thin-film transistors (TFTs) that we have transferred to flexible-polymer hosts that show a very high saturation current and transconductance. Thus very high-speed flexible electronics over large areas become possible.