Pedestrians and drivers are expected to safely navigate complex urban environments along with several non cooperating agents. Autonomous vehicles will soon replicate this capability. Each agent acquires a representation of the world from an egocentric perspective and must make decisions ensuring safety for itself and others. This requires to predict motion patterns of observed agents for a far enough future. In this paper we propose MANTRA, a model that exploits memory augmented networks to effectively predict multiple trajectories of other agents, observed from an egocentric perspective. Our model stores observations in memory and uses trained controllers to write meaningful pattern encodings and read trajectories that are most likely to occur in future. We show that our method is able to natively perform multi-modal trajectory prediction obtaining state-of-the art results on four datasets. Moreover, thanks to the non-parametric nature of the memory module, we show how once trained our system can continuously improve by ingesting novel patterns.