Explore the fascinating interplay between structure and randomness in number theory through this Presidential Lecture by Peter Sarnak from the Simons Foundation. Discover how apparent contradictions between order and chaos in mathematical systems reveal deep insights about prime numbers, arithmetic sequences, and other fundamental concepts. Learn through concrete examples how mathematicians develop models for randomness that often yield unexpected and illuminating results. Understand why identifying and proving random behavior in number-theoretic objects represents one of the most significant challenges in modern mathematics. Gain insight into how randomness serves as both a tool for understanding mathematical structures and a phenomenon that emerges naturally from deterministic systems in number theory.