What are good non-English languages for mathematicians to know? It seems that knowing French is useful if you're an algebraic geometer.  More generally, I've sometimes wished I could read German and Russian so I could read papers by great German and Russian mathematicians, but I don't know how useful this would actually be.  What non-English languages are good for a generic mathematician to know?  Are there specific languages associated to disciplines other than algebraic geometry?  
(This question is a little English-centric, but I figure it's okay because this website is run in English.)
 A: For the 19th and early 20th century literature: French, German, Russian, Italian.  Earlier, Latin.  For the future... maybe Chinese.  But it seems that new mathematics will be in English for the next decade or three at least, and after that machine translation (and hand-held cybernetic assistants, if not direct mental interface) will eliminate the need for spending lots of time learning a language.
A: Italian can be quite important if you are doing projective algebraic geometry. For example there is someone I know who would often set up a Master's project like this: take paper XXX (in Italian) and prove the main results rigorously using modern language.
A: I'm inclined to suggest Russian for dynamical systems, ergodic theory and control theory. However, a lot of the original papers have been translated by AMS to English in '50s and '60s. That is not to say that some of those translations are easy to come by, though.
A: C and LaTeX.  Speak them like it's the mother tongue.
A: Pietro Majer's comment about style reminds me of another aspect of language and translation.  When I took the basic graduate course in algebra, one of the textbooks was the second volume of van der Waerden's "Modern Algebra" in English translation.  Some years later, I got (as a present) both volumes in the original German.  I found the German much more enjoyable to read, and I don't think it was just because it was not a required textbook.  Van der Waerden's writing style is great, and I think much of it got lost in the translation.  
While I'm at it, let me mention an example where more than the style got lost.  Some years ago, I saw, on our university library's new-book shelf, a translation of Dedekind's "Was sind und was sollen die Zahlen?"  The English title was "What are numbers and what should they be?" (Better, in my opinion, would be "What are numbers and what are they for?"; probably the Germans here can come up with even better translations.)  
In general, I'm not happy to rely on translations.  Sometimes, the translators know exactly what they're doing and they do a magnificent job (example: Gödel's collected works), but too often they don't really understand the material and/or the languages, and things get garbled.  Given a choice, I'd rather read originals --- except that I've heard that translations into Russian often contain useful annotations and even additional chapters.  (Unfortunately, I don't read Russian.)
A: German, I guess, because many important languages have been developed by German and Swiss mathematicians. Words like Eigenvector, Eigenvalue, Eigenfunction, Ansatz, Nullstellensatz, Entscheidungsproblem, Vierergruppe,... and the like speak for themselves.
If you want to read Einstein, Riemann, Hilbert, Gödel, Cantor,... to name but a few you have to learn German (which is quite similar to English, by the way, because they share common roots. Both are so called Germanic languages).
A: It's worth keeping in mind that there are many levels of language competence and you don't need to be able to carry on a conversation or read great literature in a given language to read most math papers.  Personally, with a dictionary (or Google) on-hand, I can follow a French paper or book in my area just fine, albeit much more slowly than I read in English, and I've never formally studied French.  I would think the situation is similar for most well-educated Anglophones, though I think many are too intimidated to give it a try.
On the other hand, I can't even get started with Russian since I don't know the alphabet.
A: Not really an answer, but a suggestion on how to go about finding an answer while keeping a modicum of objectivity.
Although the vast majority of papers on international journals are in English, some of the journals accept papers in other languages.  Check the top journals in your favourite area and see which languages they accept.  Most of the time it's only French and German.  This may be historical or, as in the case of Springer journals, possibly geographical.
In my experience, I've found French to be the most useful language after English for contemporary papers and also, together with German, for older (say, late 19th, early 20th century) papers.  Older than that and I find that there's a problem of mathematical language, regardless the natural language into which the mathematics happens to be embedded.
Alas, my mother tongue (Spanish) has very little mathematical literature :(
Edit
I noticed the CW suggestion after I had started typing the answer.  So to restate, French has proved the most useful to me, and the topics are Differential Geometry and to some extent Lie theory.  A close second is German, on the same topics.  My main research topic, though, is Mathematical Physics and this seems to be almost all in English.
A: I'm with Deane here: I think learning foreign languages is not a very mathematically productive thing to do; of course, there are lots of good reasons to learn foreign languages, but doing mathematics is not one of them.  Not only are there few modern mathematics papers written in languages other than English, but the primary other language they are written (French) in is pretty easy to read without actually knowing it.  
Even though I've been to France several times, my spoken French mostly consists of "merci," "si vous plait," "d'accord" and some food words; I've still skimmed 100 page long papers in French without a lot of trouble.  
If nothing else, think of reading a paper in French as a good opportunity to teach Google Translate some mathematical French.
A: Just a comment, there are thousands of excellent Chinese papers written in journals which never get translated to english. Being able to read both is definitely good if you are willing to put work into researching.
[EDIT:  Douglas S. Stones]  I've been learning Chinese for roughly three-and-a-half years now and I feel it has been given an unfair treatment so far in this question.  So I will add a few reasons why I found learning Chinese has been valuable as an early mathematician.
a.  There are many talented Chinese researchers that are looking for international collaborations (here's my first: http://portal.acm.org/citation.cfm?id=1734895).
b.  Merely restating a result (that has been published only in Chinese) adds something to a paper that others can't easily match.
c.  I can pronounce Chinese names at conferences without sounding like a goose to the Chinese speakers in the audience (e.g. "Wang").  These are some of the most common surnames on the planet.
d.  In learning Chinese, you make your knowledge work for you -- you actively find papers in Chinese, etc.
e.  There was an argument about traditional vs. simplified Chinese, i.e. not being standardised.  It's pretty easy to switch between traditional and simplified (and pinyin) on a computer (I admit this becomes more complicated if you only have a hard copy or a scanned copy, but there are ways such as OCR or simply drawing the unknown character into appropriate software).  For many characters, the traditional and simplified characters are quite similar.  [Side note: If I redefined "English", "German", etc. to be "European language 1", "European language 2", etc. then "European" would not be standardised.  Why would you want to learn "European language n"?]
f.  There are native Chinese speakers everywhere!
That being said, learning Chinese (as with any language) requires a certain temperament and a long-term commitment.  But it is easier now than it has ever been in the past thanks to online learning tools.  My favourites are ChinesePod, Skritter, dict.cn, DimSum Chinese Reading Assistant (but there are plenty of others).
A: French, German, Russian. It's a pity English dominates so much.
French reads beautifully. (Also if you read older papers by
Hadamard, Stieltjes or Levy, the first thing you'll notice is their
extreme honesty (they don't use difficult terms, they
define everything, they don't try to make their arguments
appear difficult) also calculations are not condemned nor indulged with (they often just write an equation and say in words how the rest of the computation goes)).
A: My impression is that the vast majority of mathematical articles written in the last 20 years are in English and that even foreign mathematicians try to write their better papers in English. It is, for better or worse, the only way to maximize the number of people who will read your paper.
So the only real need to read in a language other than English is to be able to read older papers. Which language you need depends pretty strongly not only in which subject you are interested in but the specific topics. For algebraic geometry, French can be very useful, but Italian can be, too. In differential geometry, the two languages that arise most often are French, German, and Russian. It really depends on whose work you want to read.
Overall, French is the most common language after English, because French mathematicians have been the most reluctant to switch to writing in English. For example, it is nice to be able to read the Seminaires Bourbaki.
A: As the great mathematician Groucho Marx said, English is the lingua franca.

Ha, ha, ha. :-)
A: Hungarian. No, really!
I have read a couple of very nice expository papers in Hungarian, but that's not what I mean. Being able to speak Hungarian has been socially very helpful at conferences (I work in combinatorial number theory). Hungarians are instantly warm and trusting toward other Hungarians, and as a non-Hungarian speaker of Hungarian, I am close enough.
A: It surprised me that nobody said to learn the language of your better half... if s/he is not a native English speaker. That way, you have a better family, which is one of the precondition to have a better research life, I guess.
As a Japanese, I would say learning Japanese might help you a bit, because there are many nice math textbooks in Japanese by Kodaira, Sato, Jimbo etc. Yes some of them are translated to English but some of them have not been. 
A: If your field is complex analysis then Finnish might be your language of choice. In fact, this book talks a little about how Fred Gehring learnt Finnish in order to work on quasiconformal mappings.
A: I not a native  English  speaker  and thus my intervention here is a bit odd, to say the least. Growing up behind the Iron Curtain I learned English, but I discovered that as a  mathematician  it is more useful to learn Russian,  simply because the Soviets translated  all the influential books and articles into Russian. Often, the Russian translations would be better than the original because the translators,  influential mathematicians themselves, would correct possible errors and would add illuminating footnotes and appendices.
To this day I cannot read German, but I can read Grauert's or Brieskorn'article in Russian. Today though, English is the lingua Franca of  Science, so for a youngster it is more profitable to concentrate on Math.
A: I'd say that in general (especially for algebraic geometry, but I really do mean in general) French is probably the way to go. Sure, there've been a lot of influential German-speaking mathematicians, but before 1820 or so they mostly wrote in Latin, and many of the important papers since then are available in translation.
In my experience, though, if you just want to read mathematical papers, it's not that hard (for a fairly well-read native English speaker) to pick up the basics of either French or German, since we have so many words derived from both. Of course, if you want to be able to write or speak the language, that's a different matter, but you can get surprisingly far with a dictionary and practice. (Note: This does not help with Bourbaki or Grothendieck.)
A: A quick examination suggests that articles in French continue to appear in Annals and Inventiones with some regularity, though much more frequently in the European Inventiones than the US-based Annals. I think this quite strongly supports the position that French is a useful language to be able to read. On the other hand, neither of these two journals seems to have published anything in German, or indeed any language other than French or English, for quite some time. Germany's other major journals, Mathematische Annalen and Journal für die reine und angewandte Mathematik, also do not these days seem to publish articles in German in practice. Russian and Chinese are certainly very active mathematical languages in their countries of origin, but unlike French, major works in these languages seem typically much more subject to translation.
A: Of course it's great that today we have a scientific koiné, but I only disagree with the idea showing here and there that studying a language is a waste of time, even for the sake of studying Mathematics. Mathematics, as a human mind's product, has a cultural aspect, that of course is not relevant from the theoretical side, but that I would really regret to miss. There's a French style in doing mathematics, a German style, an Italian style, a Russian style, and so on, and being able to appreciate their features and differences is really a wonderful piece of human studies. Also, I suspect that knowing a bit how the various traditional schools work in mathematics help us a bit to learn how to think better.
Btw, as far as I knonw, one of the last important journals that published in several languages, Latin included, was the Archive for Rational Mechanics and Analysis at the times of Clifford Truesdell. 
