This grant is to support a week-long international conference in Approximation Theory. The purpose of the meeting is to bring together researchers from around the world to present their latest work in this field, and to discuss future directions for the field. Approximation theory impacts many areas of mathematics, engineering, and science. Some of its many applications include computer-aided design (especially important in the automobile, aircraft, and movie industries), computer-aided imaging (important in the medical sciences), and computational methods for modeling and analyzing engineering structures (useful for solving a variety of problems of military and security importance). The conference will bring leading senior people together with graduate students and young researchers just entering the field, with the aim of assisting in the development of young talent to bolster U.S. mathematical and scientific capabilities for the future.