After the unexpected discovery of x-rays emitted from comet C/1996 B2 (Hyakutake) with the Rontgen X-ray Satellite (ROSAT) in March 1996, x-ray emissions from comets C/1990 K1 (Levy), C/1990 N1 (Tsuchiya-Kiuchi), 45P (Honda-Mrkos-Pajdusakova), and C/1991 A2 (Arai), optically 300 to 30,000 times fainter than Hyakutake, were discovered in archival ROSAT data. These findings establish comets as a class of x-ray sources and allow their properties to be studied over a wide optical brightness range. The results indicate that charge exchange between highly charged heavy ions in the solar wind and cometary neutrals is the dominant process for the x-ray emission. Comets may thus be used as probes for monitoring the heavy-ion content of the solar wind.