Measurements of current noise have given an important experimental verification of the existence of fractional-charge quasiparticles in the quantum Hall effect. I will show that one can also use such measurements to extract information about even more exotic types of physics, such as the non-Abelian statistics of quasiparticles in a Pfaffian state (believed to characterize the u = 5/2 quantum Hall state). I will also discuss the physics of a system composed of a superconductor in contact with two nanotubes; the entangled Cooper pairs in the superconductor can be separated into the two nanotubes due to the strong electron-electron interactions in the tubes. We show that the measurement of current correlations in this system can determine whether electrons preserve their quantum entanglement upon tunneling into the two nanotubes; if so, this system could be very useful for quantum information processing. In the last part of the talk I will discuss several other possible ways to use noise measurements for understanding the physics of one-dimensional systems.