The mid-infrared (MIR), far-infrared (FIR), and terahertz (THz) frequencies are the least developed parts of the electromagnetic spectrum for applications. Traditional semiconductor technologies like laser diodes and photodetectors are successful in the visible light range but are still confronted with great challenges when extended into the MIR/FIR/THz range. In this paper, we demonstrate that topological insulators (TIs), especially those with Mexican-hat band structure (MHBS), provide a route for overcoming these challenges. The optical responses of MHBS TIs can be 1-2 orders of magnitude larger than that of normal semiconductors at the optical transition edge. We explore the databases of topological materials and discover a number of MHBS TIs whose bandgaps lie between 0.05 and 0.5 eV and possess giant gains (absorption coefficients) on the order of 104-105 cm-1 at the transition edge. These findings may significantly boost potential MIR/FIR/THz applications such as photon sources, detectors, ultrafast electro-optical devices, and quantum information technologies.