We review our experimental and theoretical studies on the ultra-small single-walled carbon nanotubes (SWNTs) fabricated in the 1 nm channels of AlPO 4 -5 (AFI) zeolite single crystals. The structure of the SWNT was characterized by transmission electron microscopy (TEM), diffuse x-ray diffraction, and micro-Raman measurements, all consistently indicating a diameter of 0.4 nm, at or close to the theoretical limit. The large curvature in the 0.4 nm SWNTs makes the nanotubes marginally stable. On the one hand, the free-standing 0.4 nm SWNTs can be thermally destroyed at a much lower temperature than larger sized SWNTs but, on the other hand, it introduces a variety of interesting material characteristics such as the large split of the G-like Raman modes, softening of the radial breathing modes, closing of the semiconducting gap so that the (5, 0) nanotubes are metallic, and the enhancement of the electron–phonon coupling that makes these ultra-small nanotubes superconducting at a relatively high temperature (15 K). Band structure and dielectric function of the 0.4 nm SWNTs were calculated using the local-density-functional approach. The calculated dielectric functions yield predictions in very good agreement with the experimentally measured absorption spectra. The absorption bands can be identified as dipole transitions between states in the vicinity of the van Hove singularities. Further confirmation of these dipole-allowed transitions was obtained by the resonant Raman excitation spectrum. Electric transport measurements were conducted on the SWNT@AFI crystals. As the zeolite matrix is insulating, electric conduction can be ascribed to the nanotubes. It was shown that the conductivity of the 0.4 nm SWNTs is governed by a 1D electron hopping process at temperatures above 20 K. The measured magnetic and transport properties revealed that at temperatures below 20 K, these ultra-small SWNTs exhibit superconducting behaviour with a mean-field superconducting transition temperature of 15 K. The superconducting characteristics display smooth temperature variations owing to 1D fluctuation. The observed anisotropic Meissner effect, the superconducting gap and fluctuation supercurrent were consistently explained on the basis of the Ginzburg–Landau formalism. By means of lithium doping, the electronic structure of the ultra-small SWNTs can be modified. Results of a first-principles calculation as well as experimental observation show that the SWNT@AFI system can adsorb lithium atoms up to a density as high as 10 wt%.