Selected highlights of more than a decade of research efforts concerning the structural chemistry of actinyl materials at the University of Notre Dame is reviewed, with an emphasis on complex topological arrangements of polyhedra to form extended structures and frameworks. Earlier work focused on structures of uranyl minerals and synthetic compounds, with increasing emphasis on neptunyl materials and the importance of cation–cation interactions in their structural details and properties. Much of the research over the past 5 years has examined a growing family of nano-scale clusters of uranyl peroxide polyhedra containing from 16 to 60 polyhedra. These clusters contain topological squares, pentagons and hexagons, and six have adopted fullerene topologies with 12 pentagons and an even number of hexagons.