The goal of an inverse analysis is the identification of unknown, potentially stochastic parameters of a system using a model of this system and experimental (and thus noisy) measurement data. Examples of unknown properties, which can be determined by means of inverse analysis, are initial and boundary conditions or material properties. Inverse problems are, from a mathematical point of view, ill-posed, i. e., the existence of a unique solution cannot be guaranteed. In combination with additional difficulties arising due to, e.g., noisy data, numerical errors, and model inadequacies, sub-optimal performance of deterministic approaches is often observed. In many cases, probabilistic approaches provide significantly more robust results and allow for calculating probability distributions for the unknown model parameters as well as a mathematically rigorous and consistent calculation of the remaining uncertainties.