The nonlinear optical response of carbon nanotubes (CNTs) to the interaction with intense ultrashort laser pulses was studied theoretically and experimentally. A full quantum-mechanical theory for harmonics generation from a single-walled CNT has been developed, using the quantum kinetic equations for π-electrons with both intraband and interband transitions taken into account. In the regime of strong driving fields, a non-perturbative approach with the numerical solution of the quantum kinetic equations in the time domain was used to calculate the density of the axial electric current in CNTs. The results of this theory are compared to experiments performed on samples of multi-walled CNTs, using pulses of 160 fs generated by a Cr:Forsterite laser, at a wavelength of 1250 nm. The experimental results show indeed an unusual nonperturbative behavior of the third-harmonic yield, for relatively low input laser fields of ~ 1010 W/cm2, in very good agreement with the theoretical predictions. The interaction of CNTs with strong laser fields results not only in the generation of harmonics, but also in the generation of a broad spectral background. Generation of a continuous background in the vicinity of the third-harmonic of the laser field was also obtained from the quantum-mechanical calculations, however, with lower intensities than observed experimentally. Possible explanations for this discrepancy will be discussed.