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In this thesis we study the interplay between electronic correlations and geometry in single-walled carbon nanotubes by microscopic model calculations. Electronic correlations are expected to be strong because of the low dimensionality of carbon nanotubes. Moreover the possibility of existing in different chiralities make them an ideal model system to investigate this interplay. After reviewing the band theory we discuss the magnitude and the scaling of the single particle charge gap when electronic correlations are included. This is done within a Hartree-Fock mean field calculation and with the help of a renormalization group argument. We predict that there is a correlation induced charge gap of several meV. This result is especially important for carbon nanotubes of armchair chirality where the band gap is always zero. We also observe that this correlation gap is tunable with uniaxial strain. In another chapter we study the electronic properties when a magnetic field parallel to the tube axis is applied. The persistent currents show a strong dependence on chirality. When we look at the diffusive limit by adding disorder (impurities) we can exhibit the Altshuler-Aronov-Spivak effect. The last two chapters are concerned with the question of superconductivity in carbon nanotubes. We calculate the spingap in the Heisenberg model for the smallest tubes by an exact quantum Monte Carlo method. We relate these results to the RVB theory of superconductivity (mean-field and variational Monte Carlo) which describes the system upon hole doping. We obtain chirality dependent RVB superconducting order parameters. Compared to the two dimensional limit we observe that superconductivity is enhanced and antiferromagnetism is reduced. Wherever possible we try to put our results into relation with experimental findings.