  We consider interpretations of the recent $\sim 3 \sigma$ reports by the CMS
and ATLAS collaborations of a possible $X(\sim 750~{\rm GeV})$ state decaying
into $\gamma \gamma$ final states. We focus on the possibilities that this is a
scalar or pseudoscalar electroweak isoscalar state produced by gluon-gluon
fusion mediated by loops of heavy fermions. We consider several models for
these fermions, including a single vector-like charge $2/3$ T quark, a doublet
of vector-like quarks $(T, B)$, and a vector-like generation of quarks, with or
without leptons that also contribute to the $X \to \gamma \gamma$ decay
amplitude. We also consider the possibility that $X(750)$ is a dark matter
mediator, with a neutral vector-like dark matter particle. These scenarios are
compatible with the present and prospective direct limits on vector-like
fermions from LHC Runs 1 and 2, as well as indirect constraints from
electroweak precision measurements, and we show that the required Yukawa-like
couplings between the $X$ particle and the heavy vector-like fermions are small
enough to be perturbative so long as the $X$ particle has dominant decay modes
into $gg$ and $\gamma \gamma$. The decays $X \to Z Z, Z \gamma$ and $W^+ W^-$
are interesting prospective signatures that may help distinguish between
different vector-like fermion scenarios.
