The insulation of modern medium or high voltage power cables typically consists of crosslinked polymers. Due to the crosslinking bonds in the polymer the maximum operating temperature can be increased. In power cables crosslinked polyethylene (XLPE) takes the largest market share. Today, a monitoring of the crosslinking process can only be achieved performing hot set tests on a sample basis at the start and end of the production. In this thesis a continuous monitoring method is developed, which can be used to detect process defects early on. Further it can provide a tool for process optimization. Hence, a continuous process monitoring can reduce scrap during production and increase overall equipment effectiveness. For the development of the monitoring method it is assumed, that ultrasonic technique can detect changes in the degree of crosslinking (DOC) of a polymer, since crosslinking changes the mechanical properties of the material. The dependence of the acoustical parameters on the DOC is not known in detail for XLPE and semiconductive cable compounds. Therefore, a comprehensive parameter study is performed using material samples in a laboratory scale. During the study the DOC, material temperature, testing frequency and wall thickness are varied. The evaluation of the measurement data reveals that all acoustical parameters show a measurable dependency on the DOC for XLPE. For distinction between different DOC using a single parameter the sound velocity can best be used. On the contrary, measurements on the semiconductive compound do not indicate a dependency of the acoustical parameters on the DOC. Single acoustical parameters exhibit only small variations dependent on the DOC. Therefore, the feasibility of multivariate classification and machine learning is investigated, to exploit the entire information contained in the ultrasonic signal. In this context different classification models are compared regarding classification accuracy. While single models can predict the degree of crosslinking with a very high precision, a sensitivity analysis shows, that most models react very sensitive to measurement errors in the input data. However, a model based on the subspace k-nearest neighbor method presents a higher robustness to input data variation. It is concluded that this model is suitable for quantitative monitoring of the DOC in an industrial environment. Furthermore, this multivariate model has the huge benefit to work without the knowledge of the material temperature. In addition, a univariate method for the qualitative monitoring of the crosslinking process of power cable insulation systems based on the sound velocity is presented. Both methods are validated in the lab using a model cable core. With the help of statistical evaluations, for the first time a continuous, metrological monitoring of the crosslinking process including recommendations for action for the line manager is made possible.