Impacts of meteoroids on the Moon should cause detectable optical flashes, but the population of objects that are big enough is very low, and hitherto no unambiguous impact flashes have been recorded. The flux of meteoroids associated with the Leonid meteor shower of 18 November 1999 was predicted to produce observable flashes on the night side of the Moon. Here we report the unambiguous detection of five such impact flashes, three of which were seen simultaneously by other observers. We also observed a possible impact flash on 16 July 1999. All of the flashes were of very brief duration (<0.02s), as expected for high-speed impacts.