We have measured the efficiencies and time responses of several large-area (200 and 80 mm2) beveled-edge avalanche photodiodes to X-rays of energy 6-20 keV in a photon counting mode. The diodes were used with avalanche gains of 10(2) to 10(3). Detector efficiency varied from approximately 50% at 8 keV to approximately 5% at 20 keV, while the time resolution was approximately 1/2 ns with a tail out to longer times (approximately 5 ns). The duration of the tail may be reduced ( < 2 ns) by selecting only larger pulses from the diode (increasing the discriminator threshold). These results are well explained by a simple physical picture of the device: the fast response is due to X-rays absorbed in the depletion layer, and the tail results from X-rays absorbed in a low-field undepleted region in front of the diode. For one device, the total active thickness (depletion and low field region) was measured to be 49 mum and the low field region to be 9 mum.