Classical learning theory is predominantly formulated under stationarity assumptions, wherein observations are drawn from a fixed probability measure. In many practical settings, however, the data-generating process evolves over time, inducing non-stationarity that fundamentally alters the limits of learnability. This paper presents an information-theoretic review of machine learning under non-stationary conditions, examining how temporal variation in the underlying distribution constrains achievable performance. We consider stochastic processes with time-indexed probability measures and analyse learning objectives in terms of excess risk, dynamic regret, and stability under distributional drift. Existing results are synthesised using information-theoretic quantities - including entropy rate, mutual information, and Kullback–Leibler divergence - to characterise how rates of change in the source process bound adaptation speed and generalisation accuracy. Across supervised learning, bandit models, and reinforcement learning, we highlight common structural dependencies between drift magnitude, information availability, and attainable error guarantees. Rather than introducing new bounds, the paper consolidates theoretical insights that reveal why persistent non-stationarity imposes irreducible performance degradation. This formulation provides a unified mathematical perspective on learning in time-varying environments and motivates the development of adaptive algorithms whose guarantees are explicitly parameterised by information-theoretic measures of change