The "boundaries" of the title refer to group action on measure spaces that enjoy the amenability property discovered by Furstenberg, Margulis and Zimmer. Over the last half-century these boundaries have been invaluable for group theory, rigidity, and more recently "bounded cohomology" -- which is the same as the cohomology of convolution algebras.

We will see that many familiar groups admit surprisingly strong ergodicity properties for boundaries. This applies to lamplighters, Thompson groups and many transformation groups. As a consequence, we determine the bounded cohomology of some of these groups.