We present the results of two Hubble Space Telescope (HST) observing programs, consisting of 11 imaging observations of 10 asteroids. The primary focus of the projects was to search for faint companions (satellites) of these asteroids. No binary systems were detected. More specifically, no companions were found at more than 0.1 arcsec separation of brightness down to 4 magnitudes fainter than the primary asteroid. No companions down to 6 magnitudes fainter than the primary asteroid were found at more than 0.5 arcsec separation. It is unlikely that companions more than 6 magnitudes fainter than a well-exposed primary would be detected. These nondetections place very stringent limits on the existence of companion bodies for these asteroids. A secondary goal was to resolve the illuminated portion of the asteroids. Nine of the 10 asteroids were marginally resolved, with three of these-9 Metis, 18 Melpomene, and 19 Fortuna-showing significant extension and brightness variations up to a factor of two across the illuminated portion of the restored image. The diameters of the resolved asteroids are generally in good agreement with those in the TRIAD II file (E. F. Tedesco, 1989, InAsteroids II(Binzel, Gehrels, and Matthews, Eds.)). Diameters for 19 Fortuna and 624 Hektor (which are not in the TRIAD file) have been measured: 225 and 370×195 km, respectively.