Mode veering is the phenomenon associated with the eigenvalue loci for a system with a variable parameter: two branches approach each other and then rapidly veer away and diverge instead of crossing. The veering is accompanied by rapid variations in the eigenvectors. In this paper, veering in structural dynamics is analyzed in general terms. First, a discrete conservative model with stiffness, mass, and/or gyroscopic coupling is considered. Rapid veering requires weak coupling: if there is instead strong coupling then there is a slow evolution of the eigenvalue loci rather than rapid veering. The uncoupled-blocked system is defined to be that where all degrees-of-freedom (DOFs) but one are blocked. The skeleton of the system is the loci of the eigenvalues of the uncoupled-blocked system as the variable parameter changes. These loci intersect at certain critical points in the parameter space. Following a perturbation analysis, veering is seen to comprise rapid changes of the eigenvalues in small regions of the parameter space around the critical points: for coupling terms of order veering occurs in a region of order around the critical points, with the rate of change of eigenvalues being of order . This is accompanied by rapid rotations in the eigenvectors. The choice of coordinates in the model and application to continuous systems is discussed. For nonconservative systems, it is seen that veering also occurs under certain circumstances. Examples of 2DOFs, multi-DOFs (MDOFs), and continuous systems are presented to illustrate the results.