The black-drop effect bedeviled attempts to determine the Astronomical Unit from the time of the transit of Venus of 1761, until dynamical determinations of the AU obviated the need for transit measurements. By studying the 1999 transit of Mercury, using observations taken from space with NASA's Transition Region and Coronal Explorer (TRACE), we have fully explained Mercury's black-drop effect, with contributions from not only the telescope's point-spread function but also the solar limb darkening. Since Mercury has no atmosphere, we have thus verified the previous understanding, often overlooked, that the black-drop effect does not necessarily correspond to the detection of an atmosphere. We continued our studies with observations of the 2004 transit of Venus with the TRACE spacecraft in orbit and with ground-based imagery from Thessaloniki, Greece. We report on preliminary reduction of those data; see http://www.transitofvenus.info for updated results. Such studies are expected to contribute to the understanding of transits of exoplanets. Though the determination of the Astronomical Unit from studies of transit of Venus has been undertaken only rarely, it was for centuries expected to be the best method. The recent 8 June 2004 transit of Venus provided an exceptionally rare opportunity to study such a transit and to determine how modern studies can explain the limitations of the historical observations.