We have calculated the static and frequency-dependent polarizability tensors of a series of (5,5)- and (9,0)-carbon nanotubes. The calculations have been performed by a dipole-dipole interaction model based on classical electrostatics and an Unsold dispersion formula. The model has previously been shown to predict successfully the frequency-dependent polarizability tensors of both aliphatic and aromatic molecules. In comparison we have carried out ab initio calculations at the Hartree-Fock level of the static polarizability of C-60, C-70, and the smaller carbon nanotubes using the STO-3G basis set. We find that the interaction model is in good agreement with the self-consistent field calculations and can be used to predict the polarizability tensors of carbon nanotubes. In addition, we find that the symmetry and intramolecular geometry of the tube have great influence on the polarizability.