Although the background level of meteors from the Leonid stream is not especially high, within the stream are dense, narrow trails of meteoroids and dust, the debris of Comet 55P/Tempel-Tuttle. These trails are similar to those discovered by the Infra-Red Astronomical Satellite in the orbits of other periodic comets. When the Earth encounters a trail, a meteor outburst or storm occurs. Two qualitatively different kinds of trail are present in the Leonid stream. `Normal' trails, up to a few centuries old, form simply because meteoroid particles of slightly smaller and larger orbital periods than the comet gradually stretch ahead of and behind the comet. The number density of particles is much higher at the centre of these trails than elsewhere in the Leonid stream. Therefore the Earth's passages close to the centre of these trails are associated with the highest Zenithal Hourly Rate storms. Trails of the second kind form as a result of resonant dynamical behaviour over more than a few centuries. Although the overall number densities of meteoroid particles in resonant trails are smaller than in normal trails, the resonant trails are rich in larger particles. This meant that the 1998 outburst had much lower ZHR than the greatest Leonid storms, but was rich in fireballs. The positions of both kinds of trail in space can be accurately calculated, by accounting for gravitational perturbations, allowing meteor storms to be predicted.