The design, modeling, fabrication, and characterization of single-photon avalanche diode detectors with an epitaxial Ge absorption region grown directly on Si are presented. At 100 K, a single-photon detection efficiency of 4% at 1310 nm wavelength was measured with a dark count rate of ~ 6 megacounts/s, resulting in the lowest reported noise-equivalent power for a Ge-on-Si single-photon avalanche diode detector (1×10-14 WHz-1/2). The first report of 1550 nm wavelength detection efficiency measurements with such a device is presented. A jitter of 300 ps was measured, and preliminary tests on after-pulsing showed only a small increase (a factor of 2) in the normalized dark count rate when the gating frequency was increased from 1 kHz to 1 MHz. These initial results suggest that optimized devices integrated on Si substrates could potentially provide performance comparable to or better than that of many commercially available discrete technologies.