AI deployed in the real-world should be capable of autonomously adapting to novelties encountered after deployment. Yet, in the field of continual learning, the reliance on novelty and labeling oracles is commonplace albeit unrealistic. This paper addresses a challenging and under-explored problem: a deployed AI agent that continuously encounters unlabeled data - which may include both unseen samples of known classes and samples from novel (unknown) classes - and must adapt to it continuously. To tackle this challenge, we propose our method COUQ "Continual Open-world Uncertainty Quantification", an iterative uncertainty estimation algorithm tailored for learning in generalized continual open-world multi-class settings. We rigorously apply and evaluate COUQ on key sub-tasks in the Continual Open-World: continual novelty detection, uncertainty guided active learning, and uncertainty guided pseudo-labeling for semi-supervised CL. We demonstrate the effectiveness of our method across multiple datasets, ablations, backbones and performance superior to state-of-the-art.