In ergodic theory, one often studies measure-preserving actions
of countable groups on probability spaces. Bernoulli shifts are a class of
such actions that are particularly simple to define, but despite several
decades of study some elementary questions about them still remain open,
such as how they are classified up to isomorphism. Progress in
understanding Bernoulli shifts has historically gone hand-in-hand with the
development of a tool known as entropy. In this talk, I will review
classical concepts and results, which apply in the case where the acting
group is amenable, and then I will discuss recent developments that are
beginning to illuminate the case of non-amenable groups.