Things to keep in mind while looking for a Postdoc overseas Hello,
I would like to receive some suggestions about what you think to be the best important things that should be kept in mind while looking for a postdoc position. I'm not considering (in this question) the opportunities one may have from direct knowledge, e.g. because they were pointed out from your advisor or in any case about positions in some nearby university where you have some contact.
For instance while it is often not written explicitly, for most positions the candidate is expected to have taken personal contacts with one professor that may be interested, and set up a research project together. But this is certainly not the only matter, and I would like to hear some story from who has some experience in this kind of issues, also for what regards things which can vary a lot depending on the country (i did my Master and I am doing Ph.D. in Italy, with a short period in France).
To make the question more focused, i'm mostly interested in the scenario of someone working in pure mathematics (e.g. number theory) making application from Europe to some university in North America, or possibly also in Asia.
(EDITED as suggested by quid, making the question CW, and more focused and about data rather than about advice).
 A: To complement Neil's answers: At Oxford and Cambridge you will find things called "Junior Research Fellowships".  These are funded by and administered by the colleges (and not the mathematics department).  They could be in any subject, or in, say, "Science and Mathematics", or even just "Mathematics" (but it's very unlikely to be more specific).  Probably the college will rotate which subject(s) it wants over the years.
These position receive a very large number applications, and so some sort of filtering system along the lines Ben suggests must happen.  Having strong references is perhaps the most important thing.
(I had one of these.  I didn't know anyone at the college before, but in the end, a mathematics fellow at the college in question worked in a field rather close to mine, and interviewed me, so presumably understood my work and was sufficiently impressed by it.  But the other postdoc in my year at the same college was in a different area, and the college simply invited in an expert from outside to interview him.  So my strong advice would be to apply for everything, and get lucky once!)
The best guide to these things remains, I think, Tom Korner's: http://www.dpmms.cam.ac.uk/~twk/fellow.pdf
I think a few other places in the UK (Warwick?  Imperial College London?) might offer one postdoc position a year under a similar format (any subject, best candidate wins)??
A: For instance while it is often not written explicitly, for most positions the candidate is expected to have taken personal contacts with one professor that may be interested, and set up a research project together.
This is not the case in the US.  Depending on exactly how one interprets what you've written, it could even be illegal to do what you've described without explicitly stating this in the job announcement.  Many postdoctoral jobs in the US are open to all areas of math; some are grant-funded and will have a more specific focus, but this will be clearly stated in the job advertisement.
What is true is that you are very unlikely to be hired for a postdoc (in the US, or I would assume anywhere else) if there isn't a professor in the department who strongly desires to hire you, and it is not so likely (though far from impossible) that this will happen because they are looking through a long list of applicants (any good postdoc will get hundreds) and just happen to look at yours.  Your prospects will improve a lot if you (or your advisor on your behalf) make contact with professors who might be interested in your research.
A: In Sheffield (which is probably typical for UK universities other than Oxford and Cambridge):


*

*Many postdoc positions are for a specific project.

*It is normal to have many applications from people that we have never heard of before.

*We would not pay much attention to teaching experience for a postdoc position. 

