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Managing inputs that are novel, unknown, or out-of-distribution is critical as an agent moves from the lab to the open world. Novelty-related problems include being tolerant to novel perturbations of the normal input, detecting when the input includes novel items, and adapting to novel inputs. While significant research has been undertaken in these areas, a noticeable gap exists in the lack of a formalized definition of novelty that transcends problem domains. As a team of researchers spanning multiple research groups and different domains, we have seen, first hand, the difficulties that arise from ill-specified novelty problems, as well as inconsistent definitions and terminology. Therefore, we present the first unified framework for formal theories of novelty and use the framework to formally define a family of novelty types. Our framework can be applied across a wide range of domains, from symbolic AI to reinforcement learning, and beyond to open world image recognition. Thus, it can be used to help kick-start new research efforts and accelerate ongoing work on these important novelty-related problems. This extended version of our AAAI 2021 paper included more details and examples in multiple domains.