Inverse problems are an expanding field with many practical applications in scientific computing and engineering. Their Bayesian enhancement encodes prior knowledge and data uncertainties into a posterior. This is an important tool in uncertainty quantification. However, performing uncertainty quantification tasks on top of this posterior is difficult to formulate and often computationally intractable. Hence, for a thesis, we propose to research the sensitivity analysis and optimal design of Bayesian inverse problems. In particular, we will first consider the scalable sensitivity analysis for the information gain in linear Bayesian inverse problems. Then, we will propose frameworks for robust optimal experimental design in non-linear Bayesian inverse problems. Extensive numerical experiments to validate and analyze the approaches will proposed and carried out. In addition, we will consider the software development necessary to reach this stage.