Each giant planet of the Solar System has two main types of moons. `Regular' moons are typically larger satellites with prograde, nearly circular orbits in the equatorial plane of their host planets at distances of several to tens of planetary radii. The `irregular' satellites (which are typically smaller) have larger orbits with significant eccentricities and inclinations. Despite these common features, Neptune's irregular satellite system, hitherto thought to consist of Triton and Nereid, has appeared unusual. Triton is as large as Pluto and is postulated to have been captured from heliocentric orbit; it traces a circular but retrograde orbit at 14 planetary radii from Neptune. Nereid, which exhibits one of the largest satellite eccentricities, is believed to have been scattered from a regular satellite orbit to its present orbit during Triton's capture. Here we report the discovery of five irregular moons of Neptune, two with prograde and three with retrograde orbits. These exceedingly faint (apparent red magnitude mR = 24.2-25.4) moons, with diameters of 30 to 50km, were presumably captured by Neptune.