  HETE-II and BeppoSAX have produced a sample of GRBs and XRFs with known
redshifts and $E_{pk}$. This sample provides four important empirical
constraints on the nature of the source jets: Log $E_{iso}$ is approximately
uniformly distributed over several orders of magnitude; the inferred prompt
energy Log $E_{\gamma}$ is narrowly distributed; the Amati relation holds
between $E_{iso}$ and $E_{pk}$; and the Ghirlanda relation holds between
$E_{\gamma}$ and $E_{pk}$.
  We explore the implications of these constraints for GRB jet structure during
the prompt emission phase. We infer the underlying angular profiles from the
first two of the above constraints assuming all jets have the same profile and
total energy, and show that such ``universal jet'' models cannot satisfy both
constraints.
  We introduce a general and efficient method for calculating relativistic
emission distributions and $E_{pk}$ distributions from jets with arbitrary
(smooth) angular jet profiles. We also exhibit explicit analytical formulas for
emission from top-hat jets (which are not smooth). We use these methods to
exhibit $E_{pk}$ and $E_{iso}$ as a function of viewing angle, for several
interesting families of GRB jet profiles. We use the same methods to calculate
expected frequency distributions of $E_{iso}$ and $E_{\gamma}$ for the same
families of models.
  We then proceed to explore the behavior of universal jet models under a range
of profile shapes and parameters, to map the extent to which these models can
conform to the above four empirical constraints.
