Single-photon detection has emerged as a method of choice for ultra-sensitive measurements of picosecond optical transients. In the short-wave infrared, semiconductor-based single-photon detectors typically exhibit relatively poor performance compared with all-silicon devices operating at shorter wavelengths. Here we show a new generation of planar germanium-on-silicon (Ge-on-Si) single-photon avalanche diode (SPAD) detectors for short-wave infrared operation. This planar geometry has enabled a significant step-change in performance, demonstrating single-photon detection efficiency of 38% at 125 K at a wavelength of 1310 nm, and a fifty-fold improvement in noise equivalent power compared with optimised mesa geometry SPADs. In comparison with InGaAs/InP devices, Ge-on-Si SPADs exhibit considerably reduced afterpulsing effects. These results, utilising the inexpensive Ge-on-Si platform, provide a route towards large arrays of efficient, high data rate Ge-on-Si SPADs for use in eye-safe automotive LIDAR and future quantum technology applications.