The study of the interaction between terahertz (THz) radiation and quantum materials has been an active area of research due to its potential for understanding fundamental physics and the development of novel technologies. In this review, we focus on the time-dependent photocurrents, behind which are the up-to-date understanding of the physical processes. We provide the recent advancements in revealing the unique properties of quantum materials via the THz emission spectroscopy. Because the theoretical interpretation of some new experimental results is still evolving, this review is intended to inspire further research in this exciting and rapidly growing field.