The Bayesian statistical method consists of updating a prior probability distribution
over the unknown parameters of a stochastic system into a posterior probability distribution
after seeing the system's output. It is perhaps the oldest statistical paradigm, going
back to the 18th century, in abstract terms as straightforward and elegant as can be, and
with the promise of not only giving a best guess of the system parameters, but also a
quantification of remaining uncertainty. Only in the last two decades has the method
been applied to infinite-dimensional parameters, most recently to inverse problems
defined e.g. by PDEs or in machine learning. We discuss some of the mathematical
issues, with a main focus on the question whether the method works and when, and
how we can define "works". We review some classical success stories and recent
findings and open questions, borrowing from our own work and that of others.

The talk will be followed by a wine reception in the Central Core CMS