This thesis contains a comprehensive presentation of fundamental problems and novel mathematical questions in structural dynamics. An introduction to the theory and a motivation of the mathematical models are given in the beginning. The basic principles of the finite element method are introduced and general remarks on ordinary differential equations are recalled, in order to simulate dynamical behavior of mechanical systems. Finally, an overview of inverse problems related to structural dynamics is given. Subsequently, the so-called model updating, the identification of load transmissions and the method of input shaping receive special attention. Although the connection between mathematics and engineering is strong, the understanding between these groups of scientists is not always easy. The text contributes to the dialog between mathematicians and engineers in the field of structural dynamics. It leads to several open problems, contains illustrating examples, as well as results from numerical and practical experiments.