Terahertz spectroscopy emerged about 13 years ago with the demonstration that nearly single-cycle pulses of far-infrared radiation could be generated, propagated through free space, and subsequently detected in the time-domain. Since then, THz spectroscopy has found widespread applicability with studies ranging from condensed matter physics to gas-phase spectroscopy to biomedical imaging. In this article, the properties and applications of THz spectroscopy are described in the context of work being done in the Schmuttenmaer labs at Yale University. In particular, it is shown that an optical pump-THz probe configuration can elucidate phenomena such as the response of low-frequency collective solvent modes in liquids, and transient photoconductivity in a variety of semiconductor systems, such as bulk GaAs, low-temperature grown GaAs, nanocrystalline colloidal TiO2, and CdSe quantum dots. In addition, recent experiments measuring charge transfer in a very direct manner are discussed.