The aim of the talk is to discuss connections between inverse problems and the emerging field of uncertainty quantification. Uncertainty quantification is necessary in inverse problems to assess statistical reliability of the obtained solutions. Ill-posedness of the underlying model generates challenges that are not typically considered in classical statistics literature. In complex parameter spaces, such as those encountered in inverse problems, calculating frequentist confidence regions can be an almost impossible task, whereas Bayesian uncertainty quantification is often computationally cheap. The problem is that the theoretical and objective meaning of such posterior based inferences is largely unclear. On the other hand frequentist uncertainty quantification is well understood and studied in traditional statistics.