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The integration of artificial intelligence (AI) and machine learning (ML) into materials science, often referred to as materials informatics or materials AI, has accelerated the discovery, design, and optimization of advanced materials. However, materials science frequently operates in small-data and sparse-regime conditions, where datasets are limited in size (often tens to hundreds of samples), high-dimensional, imbalanced, or sparsely populated due to the high cost, time, and complexity of experimental measurements and high-fidelity simulations. This narrative review synthesizes recent advances in methods tailored to these constraints, categorizing approaches at the data-source level (e.g., literature extraction, database construction, high-throughput workflows), algorithmic level (e.g., support vector machines, Gaussian process regression, ensemble models, imbalanced learning techniques), and strategic level (e.g., active learning, transfer learning). Key assumptions underlying these methods are examined, including similarity between source and target domains for transfer learning, representativeness of initial samples and reliable uncertainty quantification in active learning, and the validity of physical priors or inductive biases in physics-informed approaches. The review also addresses inherent limits, such as risks of overfitting, poor generalization beyond the training distribution, sensitivity to data quality and noise, challenges in uncertainty calibration, and dependence on domain expertise. By highlighting successful applications in property prediction, alloy design, and perovskite optimization, this work elucidates the current capabilities and boundaries of small-data and sparse-regime learning in materials AI, guiding researchers navigating data-limited environments.