Leonid storms are caused by the Earth intersecting dense trails of dust ejected from Comet 55P/Tempel-Tuttle. Here, we extend existing studies by examining the higher ejection-velocity regions of young dust trails and the circumstances around the 2031 return of 55P/Tempel-Tuttle. A model of dust trail density is successfully fitted to the observed ZHR of storms. Based on this, predictions are made for encounters in the next few years and around 2031, giving both the times and rates of maxima. The most likely prospects for encounters are from 1999--2002, especially 2001 and 2002. Details of a storm in 1869 are presented and confirm that the time of maximum is predictable to 10 minutes accuracy or better. The consequences of these findings are applied to the satellite threat and to the methods of global analysis of Leonid rates.