In this paper we report recent results on single-photon generation with single InP and CdSe quantum dots. These dots produce single-photons on demand in the visible spectral range 510-690 nm. The emitted photoluminescence was characterized by measuring the autocorrelation function and by performing Fourier spectroscopy on several transitions in the quantum dot. We present the observation and interpretation of carrier trapping and recapture in InP quantum dots which leads to anomalies in the measured correlation functions. Our interpretation is confirmed by numerical simulations based on a rate equation model.