The Kant-Laplace nebular paradigm of the origin of the Solar System fails to predict, and even inadequately interprets, observations made in the space era. In each case it has had to invoke additional hypotheses. Our paradigm considers the Solar System within present-day approach to the origin of multiple stellar systems. The Jupiter-Sun system is considered as the limiting case of a close binary, where a multiplicity of planets form on a short time scale within a rapidly rotating proto-Jupiter, a very dense analog of classical protoplanetary discs. The numerous problems associated with the origin, evolution, and manifestations of the minor bodies can be solved within the framework of their New Explosive Cosmogony. It is based on the single assumption of a buildup, in the form of a solid solution in ice, of large amounts of 2H2+O2generated in volume electrolysis of dirty ices in the massive envelopes of Ganymede-type bodies. An explosion of a body with anM<0.5Moonsheds off completely its envelope. It is such an event that happened 3.9Byrago, followed by collisional evolution, which formed apparently the MB asteroids. The explosion of a body withM > 1Moonremoves only a part of its matter in the form of water vapor and organics present in the ices, mineral particles, and large fragments of the outermost ice layers, which also contain 2H2+O2. These are the nuclei of SP comets, and their material is also capable of combustion or detonation under certain conditions. The ice explosions of Titan (only ∼104yrago), of the Galilean satellites, and of their fragments can account both for many of their properties and for the origin of the irregular satellites and Trojans, as well as for the sources of some SP and LP comets. Some of the predictions made on this basis have been confirmed.