How do you find out the latest news in fields other than your own? Most of us are at least somewhat curious about what's going on in areas of mathematics outside our own area of research.  If a significant breakthrough is made, and can be stated in language that we understand, then we would enjoy hearing about it.  Now, for truly fantastic breakthroughs like Fermat's Last Theorem or the Poincaré conjecture, the mathematical grapevine functions quite well and we all hear about it pretty quickly.  For anything below that level, however, I for one feel that there is no good way to find out what is going on in fields other than my own.  Every four years I seem to have the following conversation: "Did you hear?  So-and-so won the Fields Medal!"  "No.  And, uh, who in the world is so-and-so?"  Shameful.
What I'm wondering is, am I doing something wrong?  Does anyone else have a better way of keeping abreast, even at a superficial level, of what major advances are happening in other areas of mathematics?  At one time, Mathematical Reviews would select some articles or books for "Featured Reviews."  I really enjoyed Featured Reviews and learned about many interesting results this way.  However, MR stopped doing Featured Reviews after a while.  No official reason was given, but I have heard that one reason was that some people were treating Featured Reviews as judgments as to which papers/books were "the best," and that MR did not want to accept the responsibility for making such judgments, especially if they were going to be used for tenure and promotion decisions.  This is understandable, but unfortunate (from my point of view).
The series of books What's Happening in the Mathematical Sciences is also excellent.  However, producing an expository article of that quality takes a lot of time, and so What's Happening is necessarily limited in scope.  I would like to know what's happening in between issues of What's Happening.  If there are resources out there that others have found useful for keeping abreast of mathematical research news, I would like to hear about them.
 A: I generally subscribe to good blogs updated at a reasonable frequency from multiple fields which I think is a pretty good way to keep abreast of the news in those fields.
Here is a big list of academic blogs which might be useful. (Wayback Machine and also a link for Mathematics/Statistics)
A: This is partly redundant with the answer of Victor Protsak, but I want to emphasize his point 3. The main reason I am very happy that my library has many paper subscriptions to math journals is that every week I have a dozen issues to browse. When the title of an article rings a bell, I take a look at the abstract and, sometimes, it convinces me to read the introduction or at least the main theorems. For some journals, e.g. Annals of Math or Acta, I read all abstract regardless of the subject. It is not uncommon that I cannot understand a word, but still I cross nice or important results in a large portion of the mathematical subjects.
This is clearly not sufficient to address your question, though: I had not heard of the works of Okounkov or Werner before they get their Fields medals.
A: Here https://theorems.home.blog/theorems-list/ is the website you are asking for.
It covers all recently solved mathematical problems, which are important (for example, published in a top journal) but at the same time can be understood with not too much background.
A: It's always hard to follow what's happening, especially in fields other than your own. I don't have any silver bullets, just a few time tested things that require serious work.


*

*Talk to colleagues. If you are in a research department, always go to a colloquium. MR can be used both before and after the talk to tie it with what's been going on in the area.
Go to seminar talks.

*Conferences. Even browsing through the list of abstracts of invited addresses at AMS Sectional Meetings helps to keep you informed. Actually attending these talks and, especially, invited addresses at the Joint Math Meetings is even better: they feature people who have been doing important work recently. I imagine other Math Societies operate in a similar way. For the past few years, the JMM has run a secion on "Current events". Like the Cambridge "Perspectives in Mathematics" conference, they make the texts of all talks available. 

*Browse through new books and journals and read book reviews (the ones in the Bulletin of AMS play a role similar to the MR featured reviews, but are set in a wider context). One of the biggest draws of the JMM is the book sale. By browsing through new publications, one can get the vibes of what is happening; even better, you can probe other people's reaction to anything you find interesting but don't quite understand. Bourbaki Seminar has 18 talks a year on subjects of current research interest.

*I've heard that some people religiously follow all new submissions on the arXiv, but I can't vouch for the benefits or effectiveness of this method. On the other hand, once you get a wind of some new developments, you can search the arXiv for papers and surveys, many of which never make it to print in spite of being helpful (in some respects, arXiv now fulfills the role formerly played by the Lecture Notes in Mathematics).

*You can run a self-teach seminar if you have at least one (ideally, 3 or more) interested colleagues. This may be on the subject of a recent paper, survey, or a book. If your colleagues are in a different field, so much the better: they can teach you what's happening there, and you can teach them something in return.
