This article considers modern applied problems formalized as the Quadratic Assignment Problem (QAP) and its generalizations. The main mathematical formulations are analyzed, with particular attention to nonlinear interactions between objects, as well as modern solution methods, including exact algorithms, heuristics, and hybrid approaches. Examples of successful applications of QAP for improving the efficiency and reliability of systems in real conditions are presented. This review emphasizes the importance of developing universal and scalable optimization methods for solving complex problems.