Within the typical high school curriculum, a student has limited opportunities to acquire a full appreciation of the nature of mathematics: its wide-ranging content, the intrinsic beauty of its ideas, the nature of mathematical argument and proof, the surprising power of mathematics within the sciences and beyond. Getting a glimpse of the depth and beauty of mathematics can be a transforming experience for a student, whatever interests the student may intend to pursue in the future. The Summer Institute for Mathematics at the University of Washington is intended to provide talented, enthusiastic students with just such a glimpse. The mathematical topics studied are accessible, yet of sufficient sophistication to be challenging, allowing students to participate in the experience of mathematical inquiry and be immersed in the world of mathematics.