  One third of all HETE-2--localized bursts are X-Ray Flashes (XRFs), a class
of events first identified by Heise in which the fluence in the 2-30 keV energy
band exceeds that in the 30-400 keV energy band. We summarize recent HETE-2 and
other results on the properties of XRFs. These results show that the properties
of XRFs, X-ray-rich gamma-ray bursts (GRBs), and GRBs form a continuum, and
thus provide evidence that all three kinds of bursts are closely related
phenomena. As the most extreme burst population, XRFs provide severe
constraints on burst models and unique insights into the structure of GRB jets,
the GRB rate, and the nature of Type Ib/Ic supernovae. We briefly mention a
number of the physical models that have been proposed to explain XRFs. We then
consider two fundamentally different classes of phenomenological jet models:
universal jet models, in which it is posited that all GRBs jets are identical
and that differences in the observed properties of the bursts are due entirely
to differences in the viewing angle; and variable-opening angle jet models, in
which it is posited that GRB jets have a distribution of jet opening angles and
that differences in the observed properties of the bursts are due to
differences in the emissivity and spectra of jets having different opening
angles. We consider three shapes for the emissivity as a function of the
viewing angle theta_v from the axis of the jet: power-law, top hat (or
uniform), and Gaussian (or Fisher). We then discuss the effect of relativistic
beaming on each of these models. We show that observations can distinguish
between these various models.
