  We present results of searches for diphoton resonances produced both
inclusively and also in association with a vector boson (W or Z) using 100
pb^{-1} of p\bar p collisions using the CDF detector. We set upper limits on
the product of cross section times branching ratio for both p\bar
p\to\gamma\gamma + X and p\bar p\to\gamma\gamma + W/Z. Comparing the inclusive
production to the expectations from heavy sgoldstinos we derive limits on the
supersymmetry-breaking scale sqrt{F} in the TeV range, depending 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\to\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 couplings) of 82
GeV/c^2 at 95% confidence level.
