Over the last two decades the field of metallosupramolecular self-assembly has emerged as a promising research area for the development of intricate, three-dimensional structures of increasing complexity and functionality. The advent of this area of research has strongly benefited from design principles that considered the ligand geometry and metal coordination geometry, thus opening up routes towards rationally designed classical (Archimedean or Platonic) architectures. In this tutorial review, we will focus on more recent developments in the design and synthesis of three-dimensional suprastructures which have non-classical architectures (non-Archimedean/Platonic solids) and we will explicitly address the secondary effects responsible for their formation. Three classes of metallosupramolecular assemblies will be discussed: architectures formed through the combination of a single ligand and metal, heteroleptic structures and heterometallic structures. It is hoped that our exposition may suggest how different principles employed in these three classes of structures might be combined to create even greater complexity and potential for function.