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Summer Research, Internships, and Study Abroad

The National Science Foundation (NSF) funds several summer research experiences for undergraduates (REU) in mathematics.
The Mathematical Association of America (MAA) also maintains a list of summer REU opportunities.
The American Mathematical Society (AMS) also lists programs for students: REU's, study abroad, internships, etc.
The Joint Program in Survey Methodology (JPSM) offers paid summer internships at federal statistical agencies in Washington, D.C.
The Mathematics Advanced Study Semester (MASS) program at Penn State is a fall semester intensive program for undergraduate mathematics.
Budapest Semesters in Mathematics (BSM) offers mathematics courses (taught in English) from leading Hungarian scholars for one semester of your junior or senior year.
Math in Moscow at the Independent University of Moscow offers mathematics and computer science courses (in English) to undergraduate and graduate students.
Junior Year for Women at Smith College is a program for women in their junior year who want an intense mathematical experience (for a semester or a full year) alongside other women.
The University of Nebraska Summer IMMERSE Program is designed to help prepare students who will be entering graduate school in mathematics the following fall.