Explore the fundamental limits of computation in this thought-provoking lecture from the 2018 Ulam Lecture Series at the Santa Fe Institute. Delve into the mathematical structures that determine problem complexity, from easy to hard to impossible, as SFI Professor Cristopher Moore examines the nature of computational challenges. Discover analogies between computation and evolution, and journey from simple puzzles to universal computation, Turing's halting problem, and the essence of mathematical truth and creativity. Gain insights into how computers tackle vast solution spaces, why some problems are quickly solvable while others remain intractable, and the implications for science and society. Engage with concepts such as Hamiltonian paths, complexity classes, paradoxes, and the foundations of artificial intelligence in this comprehensive exploration of computational limits.