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Continual learning remains a long-standing challenge of machine learning. Success requires continuously ingesting new knowledge while retaining old knowledge that remains useful. In this thesis, we introduce a coherent objective for continual learning based on maximizing infinite-horizon average reward under a per-timestep computational constraint. This framing allows us to systematically reason about the design and evaluation of continual learning agents, moving beyond ad hoc metrics like accuracy retention or plasticity alone. Part I of the thesis develops foundational tools and perspectives, including an information-theoretic treatment of agent state, a quantification of information capacity, and an exploration of the stability-plasticity trade-off in continual learning. Part II presents new algorithms: a regenerative regularization method to combat plasticity loss in neural networks, Conformal Sympow--a transformer-based model that enables efficient long-context inference via learned gating and data-dependent rotations, and a diversity-driven reinforcement learning approach that enables few-shot robustness to environment perturbations. Together, these contributions help ground continual learning as a principled and tractable subfield of machine learning, bridging theory and practice.