A scattering theory for linear wave equations relates incoming and outgoing radiation that has been scattered by an obstacle. For nonlinear wave equations, waves are scattered by sources that extend into the wave zone, and lead to slow decay in time. In this talk I present joint results with Hans Lindblad, on the role of homogeneous solutions at time-like infinity, and their relation to homogeneous solutions at space-like infinity, for a scattering theory of wave equations with sources in the wave zone. These homogeneous solutions on the conformal boundary at infinity are in particular relevant in the presence of masses or charges, and I will describe the theory in the simplest setting of the linear wave equation with slowly decaying data. Homogeneous asymptotics are also relevant for kinetic equations, and in this direction I mention my recent joint work with Martin Taylor.