Omnipresent quality monitoring in food products, blood-oxygen measurement in lightweight conformal wrist bands, or data-driven automated industrial production: Innovation in many fields is being empowered by sensor technology. Specifically, organic photodetectors (OPDs) promise great advances due to their beneficial properties and low-cost production. Recent research has led to rapid improvement in all performance parameters of OPDs, which are now on-par or better than their inorganic counterparts, such as silicon or indium gallium arsenide photodetectors, in several aspects. In particular, it is possible to directly design OPDs for specific wavelengths. This makes expensive and bulky optical filters obsolete and allows for miniature detector devices. In this review, recent progress of such narrowband OPDs is systematically summarized covering all aspects from narrow-photo-absorbing materials to device architecture engineering. The recent challenges for narrowband OPDs, like achieving high responsivity, low dark current, high response speed, and good dynamic range are carefully addressed. Finally, application demonstrations covering broadband and narrowband OPDs are discussed. Importantly, several exciting research perspectives, which will stimulate further research on organic-semiconductor-based photodetectors, are pointed out at the very end of this review.