Magnetic fluxons in long Josephson junctions in many cases behave as solitons. Soliton dynamics can be experimentally studied in Josephson junctions with a great degree of precision, impossible for many other physical systems with solitons. This fact brought together the interest of mathematicians, physicists, and engineers which resulted in many theoretical and experimental studies of long Josephson junctions during the past three decades. This review describes fluxon properties and experiments with long junctions, as well as recent studies of stacked junctions (Josephson superlattices). High-frequency applications of fluxon oscillators are also discussed.