The article provides a comprehensive survey of the quadratic eigenvalue problem (QEP), covering its applications, mathematical properties, and numerical solution techniques. The QEP, which involves finding scalars \(\lambda\) and nonzero vectors \(x, y\) that satisfy \((\lambda^2 M + \lambda C + K)x = 0\) and \(y^*(\lambda^2 M + \lambda C + K) = 0\), is discussed in the context of various fields such as structural mechanics, fluid mechanics, and signal processing. The authors emphasize the importance of exploiting the structure of the matrices (dense, sparse, real, complex, Hermitian, skew-Hermitian) and the spectral properties of the problem. They classify numerical methods and catalog available software, highlighting the differences between solving the QEP in its original form and linearizing it into a generalized eigenvalue problem (GEP). The article also delves into the spectral theory of QEPs, including the Smith theorem for canonical forms and linearizations, and discusses the challenges and techniques for solving large-scale problems.The article provides a comprehensive survey of the quadratic eigenvalue problem (QEP), covering its applications, mathematical properties, and numerical solution techniques. The QEP, which involves finding scalars \(\lambda\) and nonzero vectors \(x, y\) that satisfy \((\lambda^2 M + \lambda C + K)x = 0\) and \(y^*(\lambda^2 M + \lambda C + K) = 0\), is discussed in the context of various fields such as structural mechanics, fluid mechanics, and signal processing. The authors emphasize the importance of exploiting the structure of the matrices (dense, sparse, real, complex, Hermitian, skew-Hermitian) and the spectral properties of the problem. They classify numerical methods and catalog available software, highlighting the differences between solving the QEP in its original form and linearizing it into a generalized eigenvalue problem (GEP). The article also delves into the spectral theory of QEPs, including the Smith theorem for canonical forms and linearizations, and discusses the challenges and techniques for solving large-scale problems.