Many important examples of Banach algebras are also dual Banach spaces, and over the last thirty years a theory of so-called dual Banach algebras has emerged. An interesting and important way to study a dual Banach algebra is by studying (or even classifying) its weak*-closed (left/right/two-sided) ideals. It also turns out that weak*-closed ideals have connections to other topics as well, such as asymptotic properties of group representations.

In this talk we shall give an introduction to this topic, with our main examples being the measure algebra M(G) of a locally compact group G, and algebras of convolution operators on l^p spaces. We shall describe classifications of the weak*-closed (left/right/two-sided) ideals in M(G) for certain classes of groups G. Finally, we shall outline some new results about weak*-simplicity of M(G) and algebras of convolution operators.