Consolidation stabilizes memory traces after initial encoding. Rodent studies suggest that memory consolidation depends on replay of stimulus-specific activity patterns during fast hippocampal "ripple" oscillations. Here, we measured replay in intracranial electroencephalography recordings in human epilepsy patients, and related replay to ripples. Stimulus-specific activity was identified using representational similarity analysis and then tracked during waking rest and sleep after encoding. Stimulus-specific gamma (30-90 Hz) activity during early (100-500 ms) and late (500-1200 ms) encoding is spontaneously reactivated during waking state and sleep, independent of later memory. Ripples during nREM sleep, but not during waking state, trigger replay of activity from the late time window specifically for remembered items. Ripple-triggered replay of activity from the early time window during nREM sleep is enhanced for forgotten items. These results provide the first electrophysiological evidence for replay related to memory consolidation in humans, and point to a prominent role of nREM ripple-triggered replay in consolidation processes.