This video is a lucid, dynamic presentation of the deep and important question of the relevance of mathematics to society, delivered by one of the best mathematicians of the modern age. T. Gowers gave the Keynote Address at the historic Millennium Meeting of the Clay Mathematics Institute (CMI) - the very same meeting at which the Millennium Prize Problems were announced. He spoke about the relevance of mathematics to diverse applications citing examples in computer science, finance, and engineering. His theme underscores the unity of different apparently diverse sub-fields of mathematics. His exposition is exceptionally clear and easy to follow, making mathematics accessible to non-experts. This lecture is a treasure trove of mathematical intuition and insight into the relationship between mathematics and its applications. Using historical and present-day examples, the speaker makes a convincing argument that mathematics plays a crucial role in the advancement of science.