Given an infinite matrix $M$ indexed by $\mathbb{N}_0 \times \mathbb{N}_0$, the moment problem asks whether there exists a Borel measure $\mu$ defined on the real line such that $M_{i,j} = \int x^{i+j} d\mu(x)$ for each $i,j \in \mathbb{N}_0$. If so, we say $M$ is the moment matrix for $\mu$. One can also ask whether $\mu$ is uniquely determined by $M$. In this talk, we will present results on the moment problem from the point of view of Hutchinson equilibrium measures for iterated function systems (IFSs). This is joint work with Palle Jorgensen at the University of Iowa and Karen Shuman at Grinnell College.