  Measurements of the ratio of the elastic form factors of the proton
($\mu_pG_E/G_M$) exhibit a strong discrepancy. Experiments using unpolarized
beams and Rosenbluth separation to determine the form factors have found values
of the ratio approximately consistent with unity over a wide range of $Q^2$,
while polarization transfer experiments suggest that the ratio decreases as a
function of $Q^2$. The most widely-accepted hypothesis to explain this
discrepancy is that hard two-photon exchange (TPE) significantly contributes to
the elastic $ep$ cross section. Hard TPE has been neglected in previous
analyses of electron-proton scattering scattering experiments, in part due to
the fact that there exists no model independent way to calculate the
contribution. The effect of hard TPE may be measured experimentally, however,
via precise determination of the ratio of the electron-proton and
positron-proton elastic cross sections. The OLYMPUS experiment collected more
than 3 fb$^{-1}$ of exclusive $e^- p$ and $e^+ p$ elastic scattering data at
DESY in 2012, and has determined the elastic $\sigma_{e^+p}/\sigma_{e^-p}$
ratio to unprecedented precision up to $Q^2\approx2.2$ (GeV/$c$)$^2$,
$\epsilon\approx0.4$. This presentation will discuss the OLYMPUS experiment and
analysis, and present the recently published results from OLYMPUS in the
context of the results from the other two TPE experiments.
