At the end of the 19th century, Irish astronomers Johnstone Stoney and A.M.W. Downing showed in principle how to predict meteor storms. Their work and later developments applying similar ideas are reviewed here. The theories essentially involve determining the dynamical behavior of dust trails, which are narrow, dense structures within a broader stream. The formation of dust trails and the main features of their evolution are explained. Some Leonid dust trails are perturbed by the planets so as to result in meteor storms at the start of the 21st century.