Searches for both resonant and non-resonant production of pairs of Higgs bosons ($hh$) are performed in the $b\bar{b}\gamma\gamma$ final state using $3.2$ fb$^{-1}$ of proton--proton collision data at $\sqrt{s}=13$ recorded by the ATLAS detector at the CERN Large Hadron Collider. An upper limit of $3.9$ pb on the cross-section for non-resonant production is extracted at the 95\% confidence level, while the expected limit is $5.4$ pb. In the search for a narrow $X{\to}hh$ resonance, the observed limit ranges between $7.0$ pb and $4.0$ pb for masses of the resonance between $275$ and $400$ GeV. The expected limit varies between $7.5$ and $4.4$ pb for the same mass range.

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