The late-time behavior of solutions to the wave equation on a large class of asymptotically flat spacetimes does not conform to the strong Huygens principle. Instead, it is governed by polynomially decaying "tails", as first discovered heuristically by Price. Their presence plays an important role in the study of singularities in black hole interiors. I will discuss a method for proving the precise leading-order asymptotics for the wave equation on these spacetimes and in the process I will introduce new energy decay estimates to obtain sharp decay rates that go beyond those obtained via traditional vector field methods. This talk is based on joint work with Yannis Angelopoulos and Stefanos Aretakis.