Consider the problem of exploration in sparse-reward or reward-free environments, such as Montezuma's Revenge. The curiosity-driven paradigm dictates an intuitive technique: At each step, the agent is rewarded for how much the realized outcome differs from their predicted outcome. However, using predictive error as intrinsic motivation is prone to fail in stochastic environments, as the agent may become hopelessly drawn to high-entropy areas of the state-action space, such as a noisy TV. Therefore it is important to distinguish between aspects of world dynamics that are inherently predictable (for which errors reflect epistemic uncertainty) and aspects that are inherently unpredictable (for which errors reflect aleatoric uncertainty): The former should constitute a source of intrinsic reward, whereas the latter should not. In this work, we study a natural solution derived from structural causal models of the world: Our key idea is to learn representations of the future that capture precisely the unpredictable aspects of each outcome---not any more, not any less---which we use as additional input for predictions, such that intrinsic rewards do vanish in the limit. First, we propose incorporating such hindsight representations into the agent's model to disentangle "noise" from "novelty", yielding Curiosity in Hindsight: a simple and scalable generalization of curiosity that is robust to all types of stochasticity. Second, we implement this framework as a drop-in modification of any prediction-based exploration bonus, and instantiate it for the recently introduced BYOL-Explore algorithm as a prime example, resulting in the noise-robust "BYOL-Hindsight". Third, we illustrate its behavior under various stochasticities in a grid world, and find improvements over BYOL-Explore in hard-exploration Atari games with sticky actions. Importantly, we show state-of-the-art results in exploring Montezuma's Revenge with sticky actions, while preserving performance in the non-sticky setting.