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Nanowires and nanotubes are now at the forefront of materials science at the nanoscale. This article starts with introductory comments about nanowires and nanotubes and then addresses in more detail the special structure and properties of bismuth nanowires and carbon nanotubes, which are considered as prototype examples of nanowires and nanotubes. Both nano-materials are important for the new nanoscience concepts that they introduce and for their promise for practical applications. Both provide a system that is simple enough so that detailed calculations of their properties can be carried out, and predictions about their physical behavior can be made. The occurrence and control of unusual and unique properties of specific nanostructures are the drivers for the exploitation of nanoscience in nanotechnology applications.