There is now a complete 19th-century edition of the Douay-Rheims bible on the Net. See this page.
For 20th-century translations used by Catholics, the closest thing on the Net is currently the Revised Standard Version. The full version, including deuterocanonical books (filed under "Apocrypha") can be found at the University of Michigan. In addition, Bible Gateway has the RSV, but in a version limited to the the 66-book Protestant canon.
Most passages in the on-line RSV's seem to be identical to those in the Catholic-approved editions of the RSV, with the exception of some missing books and chapters in some on-line editions. (I'd be happy to know about and point out known exceptions.)
There are a number of modern English translations that were made specifically for Catholics, such as the New American Bible and the Jerusalem Bible. These are both copyrighted. We would have to get permission to put them on the Internet. (If someone would like to look into getting permission, that could be quite helpful.)
Also, the Vulgate is available in Latin.
For the weekly scripture readings, I currently link to the RSV at gospelcom.net, since that is used in Catholic liturgies in many countries. If the RSV text for a given week is missing, though, or the on-line copy is not consistent with the Catholic version of the RSV, I'll link to the Douay-Rheims.
spok@cs.cmu.edu (Last updated 20-Jan-97)