We consider the contextual bandit problem on general action and context spaces, where the learner's rewards depend on their selected actions and an observable context. This generalizes the standard multi-armed bandit to the case where side information is available, e.g., patients' records or customers' history, which allows for personalized treatment. We focus on consistency -- vanishing regret compared to the optimal policy -- and show that for large classes of non-i.i.d. contexts, consistency can be achieved regardless of the time-invariant reward mechanism, a property known as universal consistency. Precisely, we first give necessary and sufficient conditions on the context-generating process for universal consistency to be possible. Second, we show that there always exists an algorithm that guarantees universal consistency whenever this is achievable, called an optimistically universal learning rule. Interestingly, for finite action spaces, learnable processes for universal learning are exactly the same as in the full-feedback setting of supervised learning, previously studied in the literature. In other words, learning can be performed with partial feedback without any generalization cost. The algorithms balance a trade-off between generalization (similar to structural risk minimization) and personalization (tailoring actions to specific contexts). Lastly, we consider the case of added continuity assumptions on rewards and show that these lead to universal consistency for significantly larger classes of data-generating processes.