Organic crystals with second-order optical nonlinearity feature very high and ultra-fast optical nonlinearities and are therefore attractive for various photonics applications. During the last decade, they have been found particularly attractive for terahertz (THz) photonics. This is mainly due to the very intense and ultra-broadband THz-wave generation possible with these crystals. We review recent progress and challenges in the development of organic crystalline materials for THz-wave generation and detection applications. We discuss their structure, intrinsic properties, and advantages compared to inorganic alternatives. The characteristic properties of the most widely employed organic crystals at present, such as DAST, DSTMS, OH1, HMQ-TMS, and BNA are analyzed and compared. We summarize the most important principles for THz-wave generation and detection, as well as organic THz-system configurations based on either difference-frequency generation or optical rectification. In addition, we give state-of-the-art examples of very intense and ultra-broadband THz systems that rely on organic crystals. Finally, we present some recent breakthrough demonstrations in nonlinear THz photonics enabled by very intense organic crystalline THz sources, as well as examples of THz spectroscopy and THz imaging using organic crystals as THz sources for various scientific and technological applications.