Advances in infrared astronomy and in computing power have recently opened up an interesting area of the solar system for dynamical exploration. The survey of the sky made by The Infrared Astronomical Satellite (IRAS) in 1983 revealed the complex structure of the zodiacal dust cloud. We now know the inclination and nodes of the plane of symmetry of the cloud with respect to the ecliptic and we have evidence that the cloud is not rotationally symmetric with respect to the Sun. Of even more interest is the discovery by IRAS of prominent dust bands that circle the Sun in planes near-parallel to the ecliptic. In 1984, we suggested (Dermott et al., Nature, 312, 505-509) that the solar system dust bands discovered by IRAS are produced by the gradual comminution of the asteroids in the major Hirayama asteroid families. The confirmation of this hypothesis has involved: (1) The development of a new secular perturbation theory that includes the effects of Poynting-Robertson light drag on the evolution of the dust particle orbits; (2) The production of a new high resolution Zodiacal History File by IPAC (the Infrared Processing and Analysis Center at Caltech); (3) The development of the SIMUL code: a three-dimensional numerical model that allows the calculation of the thermal flux produced by any particular distribution of dust particle orbits. SIMUL includes the effects of planetary perturbations and PR drag on the dust particle orbits and reproduces the exact viewing geometry of the IRAS telescope. We report that these tools allow us to account in detail for the observed structure of the dust bands. They also allow us to show that there is evidence in the IRAS data for the transport of asteroidal dust from the main belt to the Earth by Poynting-Robertson light drag.