In this talk, we start with basic examples of wave decay and then delve into the investigation of asymptotic expansions for both non-linear and linear wave propagation in asymptotically flat spacetimes, allowing for non-stationary spacetimes without spherical symmetry assumptions, and localized and regular initial data. We present a novel approach combining integrated local energy decay, the r^p method, and, from a spectral perspective, resolvent expansions near zero energy. Potential applications of this research include scenarios involving waves interacting with spatially-localized objects, such as solitons.