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.
Best Answer
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/