The generator problem for C*-algebras can be stated as follows: Is every separable simple unital C*-algebra singly generated? In 2014, Thiel and Winter showed that every such $\mathcal Z$-stable C*-algebra is singly generated. In this talk, we define AF-action algebras and discuss our proof that they are singly generated. We then show how this result implies that AH algebras with diagonal maps are singly generated, and in particular that a class of simple but non-$\mathcal Z$-stable algebras are singly generated, the Villadsen algebras of the first type. This talk is joint work with Chunguang Li and Zhuang Niu.