  Single-wall carbon nanotubes are almost ideal systems for the investigation
of exotic many-body effects due to non-Fermi liquid behavior of interacting
electrons in one dimension. Recent theoretical and experimental results are
reviewed with a focus on electron correlations. Starting from a microscopic
lattice model we derive an effective phase Hamiltonian for conducting
single-wall nanotubes with arbitrary chirality. The parameters of the
Hamiltonian show very weak dependence on the chiral angle, which makes the
low-energy physics of conducting nanotubes universal. The temperature-dependent
resistivity and frequency-dependent optical conductivity of nanotubes with
impurities are evaluated within the Luttinger-like model. Localization effects
are studied. In particular, we found that intra-valley and inter-valley
electron scattering can not coexist at low energies. Low-energy properties of
clean nanotubes are studied beyond the Luttinger liquid approximation. The
strongest Mott-like electron instability occurs at half filling. In the Mott
insulating phase electrons at different atomic sublattices form characteristic
bound states. The energy gaps of $0.01-0.1$ eV occur in all modes of elementary
excitations. We finally discuss observability of the Mott insulating phase in
transport experiments. The accent is made on the charge transfer from external
electrodes which results in a deviation of the electron density from
half-filling.
