How can deep learning systems flexibly reuse their knowledge? Toward this goal, we propose a new class of challenges, and a class of architectures that can solve them. The challenges are meta-mappings, which involve systematically transforming task behaviors to adapt to new tasks zero-shot. The key to achieving these challenges is representing the task being performed in such a way that this task representation is itself transformable. We therefore draw inspiration from functional programming and recent work in meta-learning to propose a class of Homoiconic Meta-Mapping (HoMM) approaches that represent data points and tasks in a shared latent space, and learn to infer transformations of that space. HoMM approaches can be applied to any type of machine learning task. We demonstrate the utility of this perspective by exhibiting zero-shot remapping of behavior to adapt to new tasks.