Artificial materials made up of atoms, molecules, and light have opened
up exciting opportunities to explore quantum physics in exotic regimes.
Through their manipulation with laser light and other tools, ultracold
gases of atoms and molecules can be used to study phenomena related to
condensed matter, high energy, and nuclear physics, and can furthermore
play host to entirely unique kinds of many-body effects. In this talk,
I'll describe how recent approaches based on the use of ``synthetic
dimensions,'' where dynamics are explored in a space relating to
internal or other auxiliary degrees of freedom, have opened up entirely
new capabilities for the study of topology and localization physics. In
one example, I'll describe how the addition of disorder to a synthetic
atomic wire led to the surprising appearance of nontrivial topological
properties. Looking forward, I'll chart out how extensions to the
``synthetic dimensions'' approach with atoms, molecules, and related
systems promise to introduce new kinds of exotic many-body states and
phenomena in the years to come.