These two talks give an introduction to Bayesian optimal experimental design (BOED) and how it relates to exploration in reinforcement learning and control. The first part deals with the theoretical foundations for BOED. Different design types are covered, e.g. space filling designs, alphabetic designs and Bayesian designs. Afterwards it is shown how Bayes' theorem relates to the mutual information (MI) or equally the expected information gain (EIG). Different estimators for estimating these quantities get explored and compared. A simple (Bayesian) linear model as well as a more complex neural network get described. The latter motivates the use of variational or amortized inference for estimating the MI - due to a very high dimensional second moment of the posterior. The inference technique is described, and it shown how it can be applied for bounding the MI. A small outlook to the second talk is given at the end.