We present results of searches for diphoton resonances produced both inclus
ively and also in association with a vector boson (W or Z) using 100 pb(-1)
of p (p) over bar collisions using the CDF detector. We set upper limits o
n the product of cross section times branching ratio for both p (p) over ba
r --> gamma gamma +X and p (p) over bar --> gamma gamma+ W/Z. Comparing the
inclusive production to the expectations from heavy sgoldstinos we derive
limits on the supersymmetry-breaking scale rootF in the TeV range, dependin
g on the sgoldstino mass and the choice of other parameters. Also. using a
NLO prediction for the associated production of a Higgs boson with a W or Z
boson, we set an upper limit on the branching ratio for H --> gamma gamma.
Finally, we set a lower limit on the mass of a "bosophilic" Higgs boson (e
.g., one which couples only to gamma, W, and Z bosons with standard model c
ouplings) of 82 GeV/c(2) at 95% confidence level.