Research Experience for Undergraduates: Summer Programs Some time ago, I found this list of REU programs held in 2009. 
The main aspects that characterize such programs are: (a) a great deal of lectures on specific topics; and, admittedly more importantly, (b) the chance to gain some hands-on experience with research projects.
I think that these programs are extremely interesting and are precious opportunities for undergraduates to gain a deeper understanding of specific mathematical topics as well as of the "work of the mathematical researcher".  
One should note, however, that most of these programs (if not all of them) are not open to European citizens (or, at least, in general non-American applicants do not receive funding).

Q: So, I would really like to hear if you know any similar programs. More
  specifically, I would like to know there are any such programs outside the U.S. (or any programs in the U.S. that accept also non-American applicants).


Remark 1: A similar question was asked on Mathematics Stack Exchange.
Remark 2: Both questions have been updated in 2015. It would be nice to receive some answers which are up-to-date. 
 A: In my experience, it depends on the specific funding of the program. Some programs funded by the NSF will require the applicants be Americans. But this is not always the case. Last year at my university, a student from Cambridge took part in the REU. 
When applying for funding for these programs, the directors must include some explanation of who will be the participants. Additionally, many of these programs specifically encourage minorities to apply, sometimes including non-American applicants.
I encourage you to look at the specific programs to see if they allow foreign applicants, emailing them if it is not stated on the webpage.
Here is the webpage that I used when searching for an REU as an undergrad.
Additionally, just googling "NSF REU 2010" is helpful.
EDIT: Shameless plug: http://www.math.ksu.edu/reu/sumar/
A: Caltech has the Summer Undergraduate Research Fellowship (SURF) program, which is open to students from non-American universities. 
I don't know what kind of Mathematics is being done at Caltech right now, but if you find some professor doing work that interests you, you can always send him an email with your CV explaining why you would like to work with him, and asking whether he will have SURF students in Summer 2011. Even if he will not, he might inform you of some colleague of his who will. The application deadline is around late February / early March 2011, I believe.The stipend is quite generous, in my opinion.
A: In the UK there is the EPSRC Vacation Bursary Scheme for undergraduates, which provides funding for students to undertake an individual research project under the supervision of a suitable advisor during the summer.  I participated as an undergrad myself and found it to be a really good experience.  As I understand it, the universities participating in the scheme deal with the organization, but there is some information available at the EPSRC website
A similar scheme exclusively for mathematics is funded by the LMS.
A: Being interested in following such a program as well, I posted a similar question on Math SE. I recently found the program held at the university of Wisconsin [which is moving to Emory, but is still being run by Ken Ono]. It specializes in number theory. As it is funded by the NSF as well, foreigners have to pay for the program themselves, but it is open to non-Americans. 
Good luck with finding a program!
A: Here is a master list of NSF REU awards in the Mathematical Sciences: REU Sites: Mathematical Sciences.  Sixty-six awards are listed.
Here is the analogous list in Computer and Information Sciences and Engineering:
CISE REU sites. The CISE director tells me (7Jan11) that 
"The 2011 sites have not been awarded and we cannot release information
until awards have been made." So the list is the current active sites that are continuing.
Perhaps the same holds for the math list.
A: As another data point, this REU for next summer clearly states that application is open to all, but that non-US citizens or green card holders cannot be compensated in the same way, due to the terms of the grant.  I suspect this is the case at most REU's, even if they don't make it as explicit.
A: Here are some summer research opportunities in the US (and one in Hong Kong) that fund non-US citizens. 
Research in Industrial Projects for Students (RIPS) 2011 at UCLA (Also has a program in HongKong)
http://www.ipam.ucla.edu/rips/
http://math.williams.edu/small/application-information/ (their website says they have funded non-US students in the past)
NIMBios 2011 Summer Program National Institute for Mathematical and Biological Synthesis (NIMBioS) at the University of Tennessee, Knoxville  http://nimbios.org/education/reu2011
DIMACS REU DIMACS/DIMATIA REU CCICADA REU Rutgers Math Department REU (foreign students enrolled at a U.S. university are eligible under the CCICADA program)  http://www.dimacs.rutgers.edu/REU/2011/announcement.html
Mathematical and Theoretical Biology Institute MTBI Summer Research Program  http://mtbi.asu.edu/SummerProgram.html
A: I thought I would also chime in for those students who consider the summer as the months between December and March and because at the University of Melbourne, we recently had to figure out if we could accept students for our Vacation Scholars Program (in the context of this question a REU) from other (Australian) schools. Thankfully our administrative staff told us there was not any red tape, provided the student could live and work in Australia, and so we are happy to take applications from outside the university. Also, at least for our department, a student showing the initiative to seek out a research environment went beyond what was immediately available at his/her university was viewed quite favourably. 
Finally, it is worth mentioning that the Australian Mathematical Sciences Institute also runs a summer school every year: 
http://ss15.amsi.org.au/
Again, ASMI summer school is really an opportunity for Australian undergrads.
A: The Fields institute in Toronto offers summer research for undergraduates
http://www.fields.utoronto.ca/programs/scientific/15-16/summer-research15/
When I was there the topics were mostly linked to the thematic program at the time and there were at least a couple of students from abroad (from Hungary in that case).
A: When I managed an REU in 2009, we weren't sure we had the funding until like May (or end of April?)... was frustrating but was the reality.  Some more highly funded programs can avoid that trouble, but depends on the university.  
