  We use Gaia DR2 to hunt for runaway stars from the Orion Nebula Cluster
(ONC). We search a region extending 45{\deg} around the ONC and out to 1 kpc to
find sources that overlapped in angular position with the cluster in the last
~10 Myr. We find ~17,000 runaway/walkaway candidates satisfy this 2D traceback
condition. Most of these are expected to be contaminants, e.g., caused by
Galactic streaming motions of stars at different distances. We thus examine six
further tests to help identify real runaways, namely: (1) possessing young
stellar object (YSO) colors and magnitudes based on Gaia optical photometry;
(2) having IR excess consistent with YSOs based on 2MASS and WISE photometry;
(3) having a high degree of optical variability; (4) having closest approach
distances well constrained to within the cluster half-mass radius; (5) having
ejection directions that avoid the main Galactic streaming contamination zone;
and (6) having a required radial velocity (RV) for 3D overlap of reasonable
magnitude (or, for the 7% of candidates with measured RVs, satisfying 3D
traceback). Thirteen sources, not previously noted as Orion members, pass all
these tests, while another twelve are similarly promising, except they are in
the main Galactic streaming contamination zone. Among these 25 ejection
candidates, ten with measured RVs pass the most restrictive 3D traceback
condition. We present full lists of runaway/walkaway candidates, estimate the
high-velocity population ejected from the ONC and discuss its implications for
cluster formation theories via comparison with numerical simulations.
