Reverse-time migration is a two-way time-domain finite-frequency technique that accurately handles the propagation of complex scattered waves and produces a band-limited representation of the subsurface structure that is conventionally assumed to be linear in the contrasts in model parameters. Because of this underlying linear single-scattering assumption, most implementations of this method do not satisfy the energy conservation principle and do not optimally use illumination and model sensitivity of multiply scattered waves. Migrating multiply scattered waves requires preserving the non-linear relation between the image and perturbation of model parameters. I modify the extrapolation of source and receiver wavefields to more accurately handle multiply scattered waves. I extend the concept of the imaging condition in order to map into the subsurface structurally coherent seismic events that correspond to the interaction of both singly and multiply scattered waves. This results in an imaging process referred to here as non-linear reverse-time migration. It includes a strategy that analyses separated contributions of singly and multiply scattered waves to a final non-linear image. The goal is to provide a tool suitable for seismic interpretation and potentially migration velocity analysis that benefits from increased illumination and sensitivity from multiply scattered seismic waves. It is noteworthy that this method can migrate internal multiples, a clear advantage for imaging challenging complex subsurface features, e.g., in salt and basalt environments. The results of synthetic seismic imaging experiments, including a subsalt imaging example, illustrate the technique.