Carbon nanotubes are small cylindrical molecules with a typical diameter of 1 nm and lengths of up to micrometers. These intriguing molecules exhibit, depending on the exact atomic structure, either semiconducting or metallic behavior. This makes them ideal candidates for possible future molecular electronics. In this thesis Janssen describes measurements on carbon nanotubes with a scanning tunnelling microscope. This instrument can measure both the atomic structure and electronic structure. The effect on the electronic structure of a kink in a nanotube, crossings of nanotubes, and the end of a nanotube is studied. Carbon nanotubes are also ideal systems to study quantization phenomena in one dimension. By cutting a nanotube to a short length the texbook particle-in-a-box model is experimentally realized. The wave functions of the electrons are made visible and the measured patterns correspond well to theoretical results. The spatial structure of the wave functions shows interference patterns which are exploited to obtain a value for an important electronic property, the Fermi velocity.