Luminous compact blue galaxies (LCBGs) are a diverse class of galaxies characterized by high luminosity, blue color, and high surface brightness that sit at the critical juncture of galaxies evolving from the blue to the red sequence. As part of our multi-wavelength survey of local LCBGs, we have been studying the HI content of these galaxies using both single-dish telescopes and interferometers. Our goals are to determine if single-dish HI observations represent a true measure of the dynamical mass of LCBGs and to look for signatures of recent interactions that may be triggering star formation in LCBGs. Our data show that while some LCBGs are undergoing interactions, many appear isolated. While all LCBGs contain HI and show signatures of rotation, the population does not lie on the Tully-Fisher relation nor can it evolve onto it. Furthermore, the HI maps of many LCBGs show signatures of dynamically hot components, suggesting that we are seeing the formation of a thick disk or spheroid in at least some LCBGs. There is good agreement between the HI and Hα kinematics for LCBGs, and both are similar in appearance to the Hα kinematics of high redshift star-forming galaxies. Our combined data suggest that star formation in LCBGs is primarily quenched by virial heating, consistent with model predictions.