Math Camps!

Mostly for high school students; some are costly but most have scholarships.

• PROMYS (Programs in Math for Young Scientists) - PROMYS is a six-week summer program at Boston University designed to encourage motivated high school students to explore the creative world of mathematics in a supportive community of peers, counselors, research mathematicians, and visiting scientists.
• SUMaC (Stanford University Math Camp) - bringing mathematically talented and motivated high-school students from across the United States, and from around the world, to Stanford University for four weeks of serious mathematical pursuits.
• MathCamp - A joint venture between the USA and Canada, this 5-week summer program rotates to a different college campus each year.
• Ross Mathematics Program - an intensive summer experience designed to encourage motivated pre-college students to explore mathematics. During those eight weeks, students are immersed in a world of mathematical discovery.
• HCSSiM (Hampshire College Summer Studies in Mathematics) - an intensive six-week residential program for enthusiastic students. Working in small classes and individually, participants will actively engage in the process of mathematical thought.
• SWIM (Summer Workshop in Mathematics) - a two week summer program at Princeton University for rising senior girls interested in mathematics.
• MathILy (serious Mathematics Infused with Levity) - Do you want to explore and create mathematics? Consider MathILy, an intensive residential summer program that emphasizes mathematics that is pure but applicable and has a core curriculum of discrete mathematics. 
• HSMC (Mathworks Honors Summer Math Camp) -  It is a six-week immersive residential program held at Texas State University in San Marcos, Texas.

To find more summer opportunities, check the AMS list of Math Summer Camps or the AWM list of Other Programs.