Recently developed InGaAs/InP devices suitable as single-photon avalanche diodes (SPADs) in the near-infrared range provide good detection efficiency and low time jitter, together with fairly low dark-count rate at moderately low temperature. However, the overall performance is still severely limited by the afterpulsing effect (due to carriers trapped in deep levels during the avalanche and later released). Experimental studies and speculations aiming to improve the overall performance are here presented. The photon detection efficiency is characterized and the primary dark-count rate is investigated, taking into account thermal generation in the InGaAs layer (absorption layer) and trap-assisted tunneling in the InP layer (multiplicat...