For Category theory there is a well known and quite active research mailing list https://www.mta.ca/~catdist/ 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?

2$\begingroup$ There certainly exist mailing lists for some areas, for example, foundations of mathematics, Banach spaces or algebraic topology. I do not know about the areas you mentioned in your post. $\endgroup$ – Martin Sleziak Apr 18 '17 at 12:03

3$\begingroup$ Doesn't this question fit better on main than here on meta? $\endgroup$ – Gerry Myerson Apr 18 '17 at 13:31

2$\begingroup$ @GerryMyerson I just moved it here because on main it was argued it fits better on meta :) $\endgroup$ – Loreno Heer Apr 18 '17 at 13:32

3$\begingroup$ Hmmm. meta is for questions about the operation of main, so I don't see it. Anyway, there's a Number Theory listserve, NMBRTHRY@LISTSERV.NODAK.EDU, but it's pretty lowvolume. $\endgroup$ – Gerry Myerson Apr 18 '17 at 13:35

$\begingroup$ The AMS once had a page where mailing lists on mathematics areas were listed. But I looked just now and did not find it. So: I guess mailing lists, although popular in the last century, are little used today. $\endgroup$ – Gerald Edgar Apr 18 '17 at 14:00

1$\begingroup$ @GerryMyerson Here is link to the question on the main: mathoverflow.net/questions/267537/… (Deleted at the moment, so visible only to 10k+ users and the OP.) $\endgroup$ – Martin Sleziak Apr 18 '17 at 14:20
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
 ALGTOPL  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)

1$\begingroup$ I did not find any archive for Real Analysis List online and I am not a subscriber. If somebody has more information about that list, please do add it to the post. (Especially knowing whether it is still active would be useful.) $\endgroup$ – Martin Sleziak Apr 19 '17 at 5:13

1$\begingroup$ I am also able to find some individual messages from category theory mailing list here but I did not find some kind of index for the whole archive from here. Also this link is dead, but it used to have some archive. $\endgroup$ – Martin Sleziak Apr 19 '17 at 5:18

2$\begingroup$ That's the thing with a mailing list: it is a mailing list, not an archive. I am a subscriber to the Real Analysis list, but have received no messages for years. So I guess it is defunct. $\endgroup$ – Gerald Edgar Apr 19 '17 at 12:15

1$\begingroup$ There's a finite group mailing list, but it's not archived on the open Web like fom, categories or ALGTOP. It's website is also hopelessly out of date. $\endgroup$ – David Roberts Apr 25 '17 at 4:53

1$\begingroup$ There's also constructivenews, groups.google.com/forum/#!forum/constructivenews for constructive mathematics of varying flavours; HoTTCafe for homotopy type theory for nonexperts (groups.google.com/forum/#!forum/hottcafe) and homotopytypetheory for experts (groups.google.com/forum/#!forum/homotopytypetheory). The finite group one is people.bath.ac.uk/masgcs/gpf.html or something like it, I think. $\endgroup$ – David Roberts Apr 25 '17 at 7:45

1$\begingroup$ @DavidRoberts Feel free to add the mailing lists which you consider interesting/useful enough. (I made the post CW as a way to encourage other users to edit it, if they have something worth adding.) $\endgroup$ – Martin Sleziak Apr 28 '17 at 18:07
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 AChanging" Paseman, 2017.04.18.

1$\begingroup$ You are attributing to me opinions which I have not expressed. I have no basis for an opinion on the number of active mailing lists in Mathematics. I limited my comment to the one mailing list with which I have some experience. $\endgroup$ – Gerry Myerson Apr 18 '17 at 22:52

1$\begingroup$ @Gerry, I apologize. I will edit the post to reflect my beliefs. Gerhard "Feel Free To Not Express" Paseman, 2017.04.18. $\endgroup$ – Gerhard Paseman Apr 18 '17 at 23:04