A persistent random walk moves at speed v for a random exponential time with mean τ before reversing. The particle density p(x,t) satisfies the telegrapher equation: τ∂²p/∂t² + ∂p/∂t = v²τ ∂²p/∂x². For small t it's a wave; for t≫τ, diffusion with D=v²τ/2. This describes run-and-tumble bacteria, photon diffusion in clouds, and correlated particle motion.