One-dimensional nanostructures, nanowires and nanotubes, represent an exciting and intellectually challenging area of research that crosses the borders between many areas of the physical sciences and engineering. Interest in these structures has been driven by fascinating issues in chemistry and physics, and the potential to impact science and technology. For example, it remains a great challenge to understand the intrinsic and potentially unique properties of nanowires and nanotubes, and thereby define new applications. This presentation will focus on addressing these critical issues. First, STM studies of the atomic structure and tunneling density of states of single-wall carbon nanotubes (SWNTs) will be described. Measurements show that SWNTs exhibit semiconducting and metallic behavior that depends predictably on helicity and diameter, and also exhibit well-defined 1D van Hove singularities. These results are compared and contrasted with theoretical calculations, and their implications discussed. Second, atomic force microscopy studies of the bending and stretching of individual nanowires and nanotubes will be discussed. The implications of these results on potential structural applications will be discussed. Lastly, the application of nanowires and nanotubes as molecular resolution, functionally-sensitive probes for chemistry and biology will be described.