Dynamic spectrum auctions have been considered a promising approach to effectively re-distribute spectrum resources in the secondary spectrum market. However, the existing spectrum auctions are limited to allocating spectrum in units of channels. Recently software defined radio technologies make exciting progress in operating radios with variable bandwidths. They push the need for designing more flexible spectrum auction frameworks that allow to allocate spectrum with variable bandwidth to the secondary user. In this paper, we design truthful spectrum auction frameworks in which secondary users can bid for, and then be actually allocated spectra with variable bandwidths. We first present a truthful framework for auctions of variable-bandwidth spectra in single collision domains, which can achieve system efficiency. Then, we propose a similar framework for multiple collision domains and rigorously show that it is also truthful. Results of extensive evaluations demonstrate that both of our spectrum auction frameworks for variable bandwidth are effective.