Given an existing trained neural network, it is often desirable to learn new capabilities without hindering performance of those already learned. Existing approaches either learn sub-optimal solutions, require joint training, or incur a substantial increment in the number of parameters for each added domain, typically as many as the original network. We propose a method called \emph{Deep Adaptation Networks} (DAN) that constrains newly learned filters to be linear combinations of existing ones. DANs precisely preserve performance on the original domain, require a fraction (typically 13\%, dependent on network architecture) of the number of parameters compared to standard fine-tuning procedures and converge in less cycles of training to a comparable or better level of performance. When coupled with standard network quantization techniques, we further reduce the parameter cost to around 3\% of the original with negligible or no loss in accuracy. The learned architecture can be controlled to switch between various learned representations, enabling a single network to solve a task from multiple different domains. We conduct extensive experiments showing the effectiveness of our method on a range of image classification tasks and explore different aspects of its behavior.