How does one find out what's happening in contemporary mathematics research? How does one find out what's happening in contemporary mathematics research? 
EDIT: I should have mentioned that I am looking for open access online sources. It so happens that I have been outside academia for quite a few years, and the gap between grad students' access to recent research (which is in the air, word of mouth + all publications immediately available) and that of a graduate who now works in an industry (few contacts if any + no access to journals) is tremendous. IMO this gap is the primary reason it's so difficult to do mathematics "on the side".
There are a few personal blogs out there, but personal blogs, even as very good ones, cannot be all-encompassing and necessarily focus on the subfields that the blogger is interested in most. There is MO, of course, but this is a Q&A site rather than a newsroom. There is word-of-a-mouth from other mathematicians, but the breadth of news one learns that way depends too much on one's contacts.
Physicists have Physical Review Focus website, which provides very accessible overviews of both recent and significant developments in Physics. This is not somebody's personal blog, but a well-organized collection of short digests by many contributors with references to the original papers. On most of the recent significant developments one gets a high level overview written by a specialist for a non-specialist, and can get access to a more detailed paper by following the supplied references.
I wonder if there is a site like that for mathematicians. Please tell if you know of one. If not, perhaps one could be built based on MO resources?
 A: An excellent resource is  the collection of  the talks given at the congresses of the International Mathematical Union.  These are available, in browsable form, online at www.mathunion.org/ICM from the first one (1893) to the last one in 2010.
A: This doesn't directly answer your question, but some universities give guest access to visitors for limited periods of time (e.g. an hour a day), so if there are any resources you really want to check out like MathSciNet or some paper whose author has not provided a arXiv version, you can read it at such an institution.
A: The question may be understood in different ways, depending on the time scale (what do you mean by "recent"? last week? last year? last three decades?) and the level at which you wish the mathematical developments to be presented (for the general public? for beginning graduate students? for professional mathematicians in general? for specialists of the field?). So the question will not have any single good answer.
Also, and here it is perhaps a difference with physics, short time scale are not very relevant in mathematics. Most of the important new ideas takes years, often decades, to be checked, developed, known, understood. So it is not a big deal if you miss the first time (or the second the third, etc.) that an important development is announced: you'll have years to catch up, and  after sufficient time the news will come from so many directions that you will not be able to miss it. This may explain why there is no equivalent of the Physical Review Focus in mathematics. 
That being said, let me discuss the resources I know. At a very short time scale,
the newsletter Headline and Deadline of an AMS  announces surprising advances (e.g. the bounded gap between primes). Mathoverflow, even though it is not its aim, also does in practice. On a longer time scale (about 5 or 10 years), one famous resource is the Bourbaki Seminar, which tries to cover important development at a specialist level. Of course, it has never been perfect in covering all mathematics, and my opinion is that it is much less good in that respect that it used to be. Yet it is a valuable resource.
A: I'm envious of the physicists for having a website like the APS website. Unfortunately in math, I think expository articles often lag far behind the original publications.
Regardless, here are some sources I use.


*

*Sign up for the ArXiv daily mailings, being specific about the
fields of your interest. Always read the abstracts (they're displayed right in the e-mail), and read the introductions of the papers that interest you. This is tough going because many papers are hard to read, but this is a pretty good way to stay up-to-date.

*Check out the ICM talks (at least their titles). Though these
are not a complete overview of modern math, and though they only
occur every four years, they'll give you enough math to think about
for more than four years.

*If you know the field or topic of your interest, try contacting a researcher directly. I do that when I don't know
about a certain field of math, and I've found we mathematicians are quite helpful to each other.
If you write an e-mail explaining your background, I bet you will
find a good number of mathematicians who are at least willing to
e-mail or Skype with you. (Just don't begin your e-mail with "I have squared the circle and proved the rationality of pi.")

*Check out videos at MSRI's website (for math in general) and the Simons Center website (for more math-physics stuff). These videos are a great resource, and workshop videos especially contain introductory overviews of contemporary topics.
A: The AMS series

What's Happening in the Mathematical Sciences, Volume 1—9

is superb, but perhaps a bit slow compared to a newspaper or blog.

 
 
 
 
 


Table of contents of Volume 9 (PDF download).
A: 1) Research statements.
Search Google for "research statement" and an area of mathematics that you are interested in.
Here's a short selection:
http://people.uwec.edu/mbirika/Aba_Mbirika_Research_Statement.pdf
http://www-personal.umich.edu/~satriano/research.pdf
http://www.math.uconn.edu/~Ji%20LI/Research-statement-JiLi.pdf
2) Online availability of prefaces, contents, supplementary material, and sometimes draft versions for new books.
Use Google advanced book search to Search for all books published in the last year (or further back as you wish) and scroll through the listings until you find something that looks new to you. Once you find a promising looking book, after trying Google preview, go directly to the publisher's page for the book and also try to find the author's home page to look for additional material.
Example: At http://books.google.com/advanced_book_search enter "mathematics" in the subject box and enter "topology" in the "Find results" box. Publication date 2014-2014. 
On the second page of results the book "Topological Signal Processing" looks new.
Publisher's page: http://www.springer.com/engineering/signals/book/978-3-642-36103-6
Author's page: http://www.drmichaelrobinson.net/research.html
A: There is the monthly journal Notices of the American Mathematical Society.
According to its site, 

(...) by publishing high-level exposition, the Notices provides
  opportunities for mathematicians and students of mathematics to find
  out what is going on in the field. Each issue contains one or two such
  expository articles that describe current developments in mathematical
  research, written by professional mathematicians. The Notices also
  carries articles on the history of mathematics, mathematics education,
  and professional issues facing mathematicians, as well as reviews of
  books, plays, movies, and other artistic and cultural works involving
  mathematics. Members keep abreast of official AMS reports, activities,
  and actions, and the news of the mathematical world, through articles
  the Notices.

A: Some conferences may post lectures online. For example, the Banff International Research Station (https://www.birs.ca/) hosts a 5 day workshop nearly every week on a contemporary topic with leading researchers, and you can watch live on their website. If you missed one you wanted to see, they also maintain a video archive for many of their talks.
(edited to include suggestions from Geoff Robinson and Yemon Choi)
There are also research lectures available from 


*

*The Mathematical Sciences Research Institute (https://www.msri.org/web/msri/online-videos),

*The Newton Institute (http://www.newton.ac.uk/webseminars),

*The Fields Institute (http://www.fields.utoronto.ca/video-archive),

*L'Institut Henri Poincaré (https://www.youtube.com/user/PoincareInstitute/videos), which contains the two latest Bourbaki seminars, as well as lectures from their thematic programs.


The first three can also be watched live.
A: Check out the programs and workshops at IPAM, http://www.ipam.ucla.edu/. IPAM is particularly strong on cross-disciplinary applied topics.
