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This book offers a self-contained presentation of the notion of stability in PDEs. The central questions of regularity and classification of stable solutions are treated at length. Specialists will find a summary of the most recent developments of the theory such as nonlocal and higher-order equations. For beginners the book walks through the fine versions of the maximum principle the standard regularity theory for linear elliptic equations and the fundamental functional inequalities commonly used in this field. The text also covers the inverse-square potential and submanifolds of Euclidean space.