Solar eclipse observations have long suggested that the white-light corona is permeated by long fine rays. By comparing photographs of the 2006 March 29 total eclipse with current-free extrapolations of photospheric field measurements and with images from the Solar and Heliospheric Observatory (SOHO), we deduce that the bulk of these linear features fall into three categories: (1) polar and low-latitude plumes that overlie small magnetic bipoles inside coronal holes, (2) helmet streamer rays that overlie large loop arcades and separate coronal holes of opposite polarity, and (3) ``pseudostreamer'' rays that overlie twin loop arcades and separate coronal holes of the same polarity. The helmet streamer rays extend outward to form the plasma sheet component of the slow solar wind, while the plumes and pseudostreamers contribute to the fast solar wind. In all three cases, the rays are formed by magnetic reconnection between closed coronal loops and adjacent open field lines. Although seemingly ubiquitous when seen projected against the sky plane, the rays are in fact rooted inside or along the boundaries of coronal holes.