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Diophantine equations or, more generally, polynomial equations whose solutions are restricted to integers, have captivated mathematicians from the era of Diophantus to modern researchers. In this review, we trace the historical evolution of solution methods from early classical techniques (such as the Euclidean algorithm and the method of infinite descent) to modern approaches involving geometry and computational methods. We survey recent advances, discuss important open problems (including exponential Diophantine equations), and examine the intersection of theory and applications in areas such as cryptography. Our exposition draws on three recent pieces of literature (2020–2025), two works from 2010–2025, and several foundational classical texts.