A number of our ongoing research efforts on carbon nanotubes will be discussed. These efforts are aimed at studying fundamental physics in nanostructures as well as developing sensing and device applications. First, we have demonstrated that individual carbon nanotube nanomechanical resonators can act as ultrasensitive inertial mass sensors with atomic-scale mass resolution. The prospects for single atomic mass unit sensitivity and chemical or isotope discrimination will be discussed. In addition, we have used DNA templates to self-assemble carbon nanotube crossbar devices with two-dimensional positional and orientational control. Transport measurements on these devices demonstrate that these solution-processed structures can be deposited intact on insulating substrates and yield device behavior. Finally, I will discuss two of our experiments probing strongly correlated electron behavior in carbon nanotubes. We have demonstrated that dilute hole systems in semiconducting carbon nanotubes are manifestations of a one-dimensional Wigner crystal. The holes form a spin chain that shows distinct regimes of magnetic ordering versus magnetic field and charge density. We also demonstrate that electrons in nominally metallic nanotubes form a one-dimensional Mott insulator, indicating that because of electron-electron interactions nanotubes are never metallic, in agreement with theoretical predictions. Our results underscore nanotubes' promise for studying a variety of tunable correlated electron phenomena in 1D.