Cognition is remarkably flexible; we are able to rapidly learn and perform many different tasks. Theoretical modeling has shown artificial neural networks trained to perform multiple tasks will re-use representations and computational components across tasks. By composing tasks from these sub-components, an agent can flexibly switch between tasks and rapidly learn new tasks. Yet, whether such compositionality is found in the brain is unknown. Here, we show the same subspaces of neural activity represent task-relevant information across multiple tasks, with each task compositionally combining these subspaces in a task-specific manner. We trained monkeys to switch between three compositionally related tasks. Neural recordings found task-relevant information about stimulus features and motor actions were represented in subspaces of neural activity that were shared across tasks. When monkeys performed a task, neural representations in the relevant shared sensory subspace were transformed to the relevant shared motor subspace. Subspaces were flexibly engaged as monkeys discovered the task in effect; their internal belief about the current task predicted the strength of representations in task-relevant subspaces. In sum, our findings suggest that the brain can flexibly perform multiple tasks by compositionally combining task-relevant neural representations across tasks.