We demonstrate that efficient meta-learning can be achieved via end-to-end training of deep neural networks with memory distributed across layers. The persistent state of this memory assumes the entire burden of guiding task adaptation. Moreover, its distributed nature is instrumental in orchestrating adaptation. Ablation experiments demonstrate that providing relevant feedback to memory units distributed across the depth of the network enables them to guide adaptation throughout the entire network. Our results show that this is a successful strategy for simplifying meta-learning -- often cast as a bi-level optimization problem -- to standard end-to-end training, while outperforming gradient-based, prototype-based, and other memory-based meta-learning strategies. Additionally, our adaptation strategy naturally handles online learning scenarios with a significant delay between observing a sample and its corresponding label -- a setting in which other approaches struggle. Adaptation via distributed memory is effective across a wide range of learning tasks, ranging from classification to online few-shot semantic segmentation.