There are more than 2000 compounds with RE/M/E 1: 1: 1 stoichiometry, where RE is a rare earth, actinide, or from groups 1 - 4, E is a late transition metal from groups 8 - 12, and E belongs to groups 13-15. In the first paper of a series, we describe the primary structural characteristics of this large class of compounds, focusing on the geometries and bonding in the [ME]"(-) anionic sublattices. We start with a detailed description of the EuAuSn lattice, by emphasizing its essential building blocks and the symmetry operations connecting them, in order to set up a way of describing systematically the geometric features of other R E M E frameworks. Nearly all these phases have eclipsed hexagonal layered arrangements (with only M-E bonds within these slabs) in the ME sublattice. In most cases, stacking the puckered layers in the third dimension gives rise to M-E, M-M, or E-E bonds, alternating in certain patterns. We point to several interesting sets of RE ME compounds, of like structure, but whose electron counts vary significantly. The structures of nonlayered networks and their most common electron counts are also introduced.