"Industry"/Government jobs for mathematicians Suppose that you graduate with a good PhD in mathematics, but don't necessarily want to go into academia, with the post-doc years that this entails. Are there any other options for continuing to do "real math" professionally?
For example, how about working at the NSA? I don't know much of what is done there -- is it research mathematics? Are there other similar organizations? Perhaps corporations that contract with the federal government? Companies like RSA?
Other areas of industry? Is there research mathematics done in any sort of financial or tech company?
I've made this a community wiki, since there aren't any right answers...
 A: Having worked briefly in the defense contracting industry, I concur with the other commenters. One thing worth pointing out though is that unlike with things like NSF grants, the overall standard to get government money for defense contractors is pretty low, and you get a lot of "bogosity" in many contracts awarded, including those that are funding PhD's in math-related areas who are supposedly doing research-type activities. As a result there is a really big difference in the type of work that goes on at say the NSA or the IDA compared to many other places. I saw projects that got funded for years that had very obvious flaws in their models that most of the posters here would easily point out. Even if companies eventually figure it out they are sort of stuck pursuing their old methods because that's what will bring in government money, not to mention what conceding they're wrong would do to their reputations.
Moral of the story: there is interesting math of an applied bent out there in the defense industry, but make sure you know what you're getting into. 
A: In the United States, it is definitely possible to do mathematical research at some industry labs, and publish/collaborate just as an academic would. (This is much easier if your research straddles math and computer science, but still possible in some other cases.) My employer, IBM Research, has labs in the New York and San Francisco areas which hire mathematicians to do research with a strong computer science slant. Also, Microsoft Research in Redmond, WA, Mountain View, CA, and Cambridge, MA (and the other Cambridge!) hires people with similar background. Google and Yahoo also have labs but I am not sure they do as much theoretical work. 
In the San Francisco area, there are also government-affiliated labs like Sandia, Lawrence Berkeley, and Lawrence Livermore. However I am not sure how much of their work they can publish, and what kind of security clearance they need to work on interesting projects. (They do publish at least some papers, though.)
Every group has different foci, so your chances of working in industry really depend on how much your interests and work intersect with some group. It will require some effort on your part to find appropriate places for you.
Note the above groups are just those that I am personally familiar with. Also, the above labs hire postdocs as well (in fact I am currently one). 
A: If you do not want to spend many years in postdocs, do not mind to move in another country, and speak french fluently you can try to get hired in a french university. Some people get permanent positions right after the thesis, and most of the hired people defended their phD only one or two years before. The competition is harsh, but the postdocs are not mandatory.
Edit in 2017: in the past years, among other things the importation of the postdoc system to France has changed things. Now it is rarer (though possible) to be hired just after PhD, and the competition has become even harsher. Salaries have not increased, so I guess the attractiveness for US applicants might be quite low.
A: I would browse some journals in your particular research area to see if any of the authors work for a non-academic institution.   
In my experience most of the research work done in industry/gov't is considered properitary or non-disclosable.   So there is not much opportunity to publish results.
A: I have worked in academia, at the research center of a telecommunications company (Tellabs), and at two different FFRDCs (MIT Lincoln Laboratory and IDA).  At all of the non-academic jobs, I have done "real math," published papers, attended conferences, given talks, etc.  So it is certainly possible to continue doing "real math" outside of academia.
You should be aware, however, that in almost any non-academic job, there is pressure on you to produce results that are "useful" for the company or the government.  The amount of such pressure varies, but it always exists, because ultimately that is the main justification for your paycheck.  In academia, the corresponding fact is that in almost any academic job, there is pressure on you to teach, since that is usually the justification for a significant portion of your salary.  Finding a non-academic job where there is no pressure on you to do anything "useful" is akin to finding an academic job where you have no teaching responsibilities.
Certain high-tech companies and certain FFRDC's recognize that a good way to attract top talent is to give their employees the freedom to pursue their own research interests, whatever that may be.  All the non-academic jobs I had were like this.  They actively encouraged me to spend some amount of my time doing "real math" regardless of whether the results were of any "use."  How much time?  Well, if the company was doing well, and if I was doing a good job of producing "useful" results that they liked, then they would give me more freedom.  But if the company was doing poorly then they would start to squeeze.  During the telecom industry meltdown in the late 1990s, Tellabs eventually eliminated its research center entirely, along with my job; Bell Labs (more famously) suffered a similar fate.
So far I have been drawing a dichotomy between "what the company finds useful" and "real math," and maybe you don't find that satisfactory.  After all, if you're sufficiently motivated, you can do "real math" on your own time regardless of what your "day job" is.  Maybe what you want is a job where providing what is useful to the company involves doing real math.  This is a taller order; for example, at Lincoln Labs I found that there was almost no real math involved in the work they wanted me to do, and I eventually left that job for that reason even though it was a great job in almost every other respect.  However, it is still possible to find such jobs, depending on what area of math you are interested in.  If you are interested in large cardinals and are hoping for a job where your theorems about large cardinals will be "useful" then you are probably out of luck.  However, if your interests lean towards areas with known relevance to computer science or various branches of engineering then your chances are much better.  The NSA scores pretty well in this regard since it is no secret that number theory and various other branches of so-called "pure" mathematics are relevant to cryptology.
In summary, jobs where you do "real math" do exist.  When considering such a job, though, you should first ask yourself, will I enjoy producing what this company considers to be "useful" results?  If the answer is no, then you will probably not be happy at the job even if they give you some freedom to do "real math."  However, if the answer is yes, and the company gives you some amount of freedom to do "real math," then it will probably be an excellent fit for you.
A: Defense/aerospace, government, and finance (perhaps less so now) are all industries that engage mathematicians. I have personally worked in defense and defense-related industry for my entire (still short) career (think tank, defense university, R&D), and found a great deal of intellectual freedom and interesting work in every job I've had.
The Agency mathematicians with PhDs I've met over the last decade or so are clearly very competent. While I won't speculate on what they do on a daily basis, the people I have met uniformly seemed to be engaged in activity that they found interesting (though at least two have since left to other government posts). They go through an introductory program to teach them relevant techniques and rotate through various offices for a few years to learn the lay of the land. NSA also has affiliated FFRDCs that support internal mathematical research. I have met people affiliated with these as well and they all seem to enjoy the work and to find it fulfilling. 
As an aside or two, the field of symbolic dynamics was basically inaugurated by Hedlund while working at one of these FFRDCs, and James Simons (of Chern-Simons and hedge fund fame) is also a well known alumnus. There are many other prominent mathematicians who have worked at or continue to consult for these centers, and if you search CVs you can probably find quite a few examples. 
Finally, regarding the Agency, I would suggest looking at this.
Regarding other defense/aerospace work, you're likely to find things that involve a lot of DSP or probability (or both). 
In finance, there will be a heavy emphasis on probability, particularly stochastic differential equations and martingales. I haven't had any direct experience with or inclination to try to land a spot in this field, but I suspect that (given the pay) it is quite selective. Certainly you'll need to program (probably in C).
