Single-walled carbon nanotubes, tubular crystals of sp2-bonded carbon atoms that are just one atom thick, come in different varieties, each with a subtle difference in structure and properties – some of them are metals and others are semiconductors. This talk will describe magneto-optical studies on carbon nanotubes in high magnetic fields, which confirm theoretical predictions that the band structure of a carbon nanotube is dependent on the magnetic flux threading the tube, in a truly unique manner. We observed significant field-induced optical anisotropy as well as red shifts and splittings of absorption and photoluminescence peaks. The amounts of shifts and splittings depend on the value of the magnetic flux and are quantitatively consistent with theories based on the Aharonov-Bohm effect. This represents the first evidence of the influence of the Aharonov-Bohm phase on the band gap of a solid. These results further imply that in even stronger magnetic fields the gap can disappear altogether, causing the semiconducting nanotubes to transform into metals.