Nowadays, mathematical optimization plays an important role in many applications. Often not only one objective is desired to be optimized but several aims have to be considered at the same time. For example, in manufacturing not only costs should be minimized but at the same time the product should be of high quality. The development of theoretic and algorithmic principles for the mathematical description of these problems is the concern of multiobjective optimization. In the present thesis, different multiobjective optimization problems from mechanical engineering are studied. Motivated by the example of the operating point assignment of a linear drive, the focus of this thesis lies on the study of parametric multiobjective optimization problems. Firstly, a new approach is presented which allows to solve time-dependent multiobjective optimization problems by use of numerical path following algorithms. Subsequently, the computation of solutions which change as little as possible under variations of the external parameter is another central topic of this thesis. For the numerical approximation of these so-called robust Pareto points, in this work two new approaches are presented. The first approach is based on the classical calculus of variations while the second one makes use of numerical path following methods. Finally, geometrical properties of the solution set of non-convex, parametric multiobjective optimization problems are studied and connections to bifurcation theory are pointed out.