P=?np

In 1955, John Nash sent a remarkable letter to the National Security Agency, in which— seeking to build theoretical foundations for cryptography—he all but formulated what today we call the P ? = NP problem, considered one of the great open problems of science. Here I survey the status of this problem in 2017, for a broad audience of mathematicians, scientists, and engineers. I offer a personal perspective on what it’s about, why it’s important, why it’s reasonable to conjecture that P ̸= NP is both true and provable, why proving it is so hard, the landscape of related problems, and crucially, what progress has been made in the last halfcentury toward solving those problems. The discussion of progress includes diagonalization and circuit lower bounds; the relativization, algebrization, and natural proofs barriers; and the recent works of Ryan Williams and Ketan Mulmuley, which (in different ways) hint at a duality between impossibility proofs and algorithms.