Data-driven approaches to modeling physical systems fail to generalize to unseen systems that share the same general dynamics with the learning domain, but correspond to different physical contexts. We propose a new framework for this key problem, context-informed dynamics adaptation (CoDA), which takes into account the distributional shift across systems for fast and efficient adaptation to new dynamics. CoDA leverages multiple environments, each associated to a different dynamic, and learns to condition the dynamics model on contextual parameters, specific to each environment. The conditioning is performed via a hypernetwork, learned jointly with a context vector from observed data. The proposed formulation constrains the search hypothesis space to foster fast adaptation and better generalization across environments. We theoretically motivate our approach and show state-of-the-art generalization results on a set of nonlinear dynamics, representative of a variety of application domains. We also show, on these systems, that new system parameters can be inferred from context vectors with minimal supervision.