The weak form quadrature element method (QEM) combines the generality of the finite element method (FEM) with the accuracy of spectral techniques and thus has been projected by its proponents as a potential alternative to the conventional finite element method. The progression on the QEM and its applications is clear from past research, but this has been scattered over many papers. This paper presents a state-of-the-art review of the QEM employed to analyze a variety of problems in science and engineering, which should be of general interest to the community of the computational mechanics. The difference between the weak form quadrature element method (WQEM) and the time domain spectral element method (SEM) is clarified. The review is carried out with an emphasis to present static, buckling, free vibration, and dynamic analysis of structural members and structures by the QEM. A subroutine to compute abscissas and weights in Gauss–Lobatto–Legendre (GLL) quadrature is provided in the Appendix.