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 pp¯ collisions using the CDF detector. We set upper limits on the product of cross section times branching ratio for both pp¯→γγ+X and pp¯→γγ+W/Z. Comparing the inclusive production to the expectations from heavy sgoldstinos we derive limits on the supersymmetry-breaking scale 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 H⃗ γγ. Finally, we set a lower limit on the mass of a “bosophilic” Higgs boson (e.g., one which couples only to γ, W, and Z bosons with standard model couplings) of 82 GeV/c2 at 95% confidence level.