Summer research programs are a great way of getting involved in mathematical research, working on a significant problem with a group of other students and faculty and living in a different part of the country for a while. There are many different programs offering you the opportunity to participate in some mathematical research during the summer. The programs vary widely in size, orientation, and goals. The best thing to do is to look around and try to find a few programs that match your desires as closely as possible. Don't forget that there are many similar programs in other disciplines. A brief description of some of the more important programs in mathematics is given below. Deadlines for these programs tend to be in February. You will want to get started on your research in plenty of time.
 

• The single largest program is the NSF-sponsored Research Experiences for Undergraduates (REU) program. Each year, the NSF supports many different programs across the country in all areas it funds. Although the details vary with the specific program, most of them involve a small group of undergraduates for 8 to 10 weeks working together on some research problem. You get some travel expenses and a modest stipend. Last year there were programs in many areas of mathematics, including numerical analysis, probability, geometry, discrete mathematics, fractals, parallel computing and nonlinear dynamics, not to mention REU's in other disciplines.  A brief description is given on the NSF website.  The NSF has a search page to help you find a suitable program.  The Mathematical Sciences list has more than 50 programs in many areas.  The American Mathematical Society also has an REU page providing brief descriptions of REU programs and individual links.
  

• IAS/PARK CITY MATHEMATICS INSTITUTE. This is a very interesting program bringing together research mathematicians, graduate and undergraduate students, college and high school teachers to work for three weeks on a single topic from a variety of perspectives. The 2007 topic was Statistical Mechanics; the topic for 2008 is Analytic and Algebraic Geometry. The summer school program is described here, and there are links to forms and more information.  Application deadline is January 20.
  

• One successful program aimed at freshmen and sophomores is the Carleton Summer Mathematics Program for Women. This is a four-week program emphasizing the development of mathematical research skills on topics not usually available in the undergraduate curriculum. Only open to women. Application deadline is February 22.            
  

• Another successful program aimed at women is the Edge program.   This year based at Pomona College, it will involve core courses in analysis and algebra, mincourses on various topics and guest visits and lectures.  Applicants should be women graduating seniors.  Applcation deadline is March 1.
  

• Carnegie Mellon has a seven week applied mathematics program for sophomores and juniors. Students work on research projects and take a class in analysis.  The stipend is $2250. Application deadline is March 1.
  

• The National Security Agency has a very highly-regarded 12-week summer program introducing students to various areas of cryptomathematics.  You need to be a US citizen and get a security clearance.  Application deadline is October 15.
  

• The Mathematical Association of America (MAA) has a student page, which includes links to many of these and other programs, as well as having information on careers, conferences and competitions.