  Recent advances in astronomical instrumentation mean that we are now able to
image the thermal emission from the disks of dust around main sequence stars
that may be the fossil remnants of planetary formation. These observations
imply that the distribution of dust in the debris disks is neither smooth nor
symmetrical; e.g., mid-infrared images of the disk of dust around the young A0V
star HR 4796A show two lobes of emission, one of which may be ~5% brighter than
the other. The observed structure of the debris disk in the solar system, i.e.,
the zodiacal cloud, also contains asymmetries: it has an offset center of
symmetry, it is warped, and there is an asymmetric ring of dust co-orbiting
with the Earth. Since the zodiacal cloud's asymmetries have been shown to be
signatures of the gravitational perturbations of the solar system's planets, it
is hoped that it may be possible to indirectly detect extrasolar planetary
systems by their signatures in debris disk observations. This dissertation uses
the physical processes that affect the evolution of debris material in the
solar system to create a generalized model for the evolution of circumstellar
debris material. It then shows how planetary perturbations affect that
evolution, thereby causing the signatures of planets seen in the structure of
the zodiacal cloud. This model can be used to provide a quantitative
interpretation of debris disk observations, and the necessary modeling
techniques are demonstrated by their application to observations of the HR 4796
disk. As well as determining the large scale structure of the HR 4796 disk, the
modeling shows how a small body (>10 Earth mass) in the HR 4796 system that is
on an orbit with an eccentricity larger than 0.02 could be the cause of the
observed brightness asymmetry.
