Single-photon detectors (SPDs) are the most sensitive instruments for light detection. In the near-infrared range, SPDs based on III-V compound semiconductor avalanche photodiodes have been extensively used during the past two decades for diverse applications due to their advantages in practicality including small size, low cost and easy operation. In the past decade, the rapid developments and increasing demands in quantum information science have served as key drivers to improve the device performance of single-photon avalanche diodes and to invent new avalanche quenching techniques. This Review aims to introduce the technology advances of InGaAs/InP single-photon detector systems in the telecom wavelengths and the relevant quantum communication applications, and particularly to highlight recent emerging techniques such as high-frequency gating at GHz rates and free-running operation using negative-feedback avalanche diodes. Future perspectives of both the devices and quenching techniques are summarized.