Recent years have witnessed an increased demand for more powerful computers and continuous miniaturization of electronics. As the drive toward miniaturisation of electronic devices continues, the utilization of the one-dimensional electronic properties would facilitate very fast and functional devices. A system that provides a unique template for studying the one-dimensional electronic behaviour is present in carbon nanotubes, which have been proposed as nanoscale building blocks because of their exceptional physical properties, making them an ideal candidate for future solid-state nanoelectronic devices. For the nanotubes to be part of a larger assembly, their properties, together with their response to different environments need to be examined. An ideal tool for probing tiny structures such as carbon nanotubes is the scanning tunnelling microscope, which has the ability to obtain a direct real space image with extremely high spatial resolution, allowing also investigations of the electron distribution in a material. Measurements performed with the scanning tunnelling microscope of the electronic states of carbon nanotubes are reported and several key electronic phenomena are demonstrated in this study. The explicit relationship between the atomically resolved structure and the corresponding electronic behaviour of carbon nanotubes, as predicted by theory for carbon nanotubes, is revealed by our measurements, which are further validated by calculations. Evidence indicative of interlayer interaction between the two constituent shells of double walled carbon nanotubes is found and the chirality of the inner tube determined, based on a detailed analysis of the tunnelling spectra. Our experiments indicate that the overall electronic structure is dependent on the interlayer interaction and not only on the chirality pairs as was previously suggested. Radial deformations of carbon nanotubes have been studied and found to drastically affect the electronic properties of carbon nanotubes, hi particular, the effect on the electronic properties of a twist-induced deformation was observed to be reversed by the radial collapse of the tube. Due to their dimensions and geometry, carbon nanotubes also provide a unique opportunity of engineering novel one-dimensional structures within their cores. Once successfully introduced into the nanotubes, the effect that the encapsulated filling material (AgNO3, AgI and HgTe) exerts on the encasing nanotube was investigated by scanning tunnelling microscopy. The results are discussed in connection with some of the encountered difficulties that sometimes precluded reliable measurements.