  The OLYMPUS experiment was designed to measure the ratio between the
positron-proton and electron-proton elastic scattering cross sections, with the
goal of determining the contribution of two-photon exchange to the elastic
cross section. Two-photon exchange might resolve the discrepancy between
measurements of the proton form factor ratio, $\mu_p G^p_E/G^p_M$, made using
polarization techniques and those made in unpolarized experiments. OLYMPUS
operated on the DORIS storage ring at DESY, alternating between 2.01~GeV
electron and positron beams incident on an internal hydrogen gas target. The
experiment used a toroidal magnetic spectrometer instrumented with drift
chambers and time-of-flight detectors to measure rates for elastic scattering
over the polar angular range of approximately $25^\circ$--$75^\circ$. Symmetric
M{\o}ller/Bhabha calorimeters at $1.29^\circ$ and telescopes of GEM and MWPC
detectors at $12^\circ$ served as luminosity monitors. A total luminosity of
approximately 4.5~fb$^{-1}$ was collected over two running periods in 2012.
This paper provides details on the accelerator, target, detectors, and
operation of the experiment.
