For Category theory there is a well known and quite active research mailing list https://www.mta.ca/~cat-dist/ Do similar mailing lists or usenet groups exist for other branches of mathematics? For example in metric geometry, conformal geometry,... Or do most research related discussions now take place on MO?
Here is a list of some mailing list for specific areas of mathematics. (This is posted as a community wiki, if you are aware of some other mailings lists, do not hesitate to add them to the list. And also if you have some additional useful information related to some of the lists.)
- FOM -- Foundations of Mathematics - basic info, archive. This mailing list has been mentioned quite often in MO posts and elsewhere, including citations in some books and papers.
- Banach -- Banach Space Theory News - basic info, archive
- ALGTOP-L -- Algebraic Topology Discussion Group - basic info, archive
- Categories List (category theory) - basic info, archive until 2009
Real Analysis List- basic info (now very likely defunct - see GEdgar's comment)
Sci.math, sci.math.research, alt.sci.(whatever) are some of the newsgroups that dealt with mathematics and various branches. You can find histories of them and other information on them with web searches.
The unmoderated groups became spam wastelands, especially after Google instituted their version called Google Groups. I left sci.math for this reason, and only looked sparingly at sci.math.research, which was moderated. When Kevin Buzzard posted in s.m.r. about MathOverflow back in 2010, I tried it, and have not looked back since.
If you frequent some older web pages (using the Wayback Machine, for example), you will find references to mailing lists and other resources. It seems that blogs and forums and YouTube are the current means of idea exchange; (as I believe Gerry Myerson might say) I will be surprised if you find many active mailing lists.
Gerhard "The Technologies, They Are A-Changing" Paseman, 2017.04.18.