We present results from three deep ROSAT high-resolution imager observations of the Orion Nebula star-forming region. The X-ray images contain over 1500 catalogued stars in a roughly 0.8deg^2^ region centered on the Trapezium. In all, 389 distinct X-ray sources have been detected, at least two-thirds of which are associated with a single proper-motion cluster member. X-ray emission is detected from stars of all spectral types, from the massive O- and B-type components of the Trapezium to the coolest, low-mass pre-main-sequence (PMS) stars. In this paper, we focus primarily on X-ray emission from the late-type PMS stars. Of the ~100 late-type cluster members with measured spectral types, approximately three-fourths have been detected; we have derived X-ray luminosity upper limits for the remaining stars. We found coronal X-ray emission turns on around spectral type F6, with the upper envelope of activity increasing with decreasing effective temperature.