Optical quantum information science and technologies require the capability to generate, control, and detect single or multiple quanta of light. The need to detect individual photons has motivated the development of a variety of novel and refined single-photon detectors (SPDs) with enhanced detector performance. Superconducting nanowire single-photon detectors (SNSPDs) and single-photon avalanche diodes (SPADs) are the top-performer in this field, but alternative promising and innovative devices are emerging. In this review article, we discuss the current state-of-the-art of one such alternative device capable of single-photon counting: the resonant tunneling diode (RTD) single-photon detector. Due to their peculiar photodetection mechanism and current-voltage characteristic with a region of negative differential conductance, RTD single-photon detectors provide, theoretically, several advantages over conventional SPDs, such as an inherently deadtime-free photon-number resolution at elevated temperatures, while offering low dark counts, a low timing jitter, and multiple photon detection modes. This review article brings together our previous studies and current experimental results. We focus on the current limitations of RTD-SPDs and provide detailed design and parameter variations to be potentially employed in next-generation RTD-SPD to improve the figure of merits of these alternative single-photon counting devices. The single-photon detection capability of RTDs without quantum dots is shown.