The bismuth polyhedra in ternary transition metal-centered bismuth cluster halides may form discrete molecules or ions, infinite chains, and/or infinite layers. The chemical bonding in many of these diverse structures is related to that in deltahedral boranes exhibiting three-dimensional aromaticity by replacing the multicenter core bond in the boranes with two-center two-electron (2c-2e) bonds from the central transition metal to the nearest neighbor bismuth vertices. Examples of discrete molecules or ions include octahedral MBi6(mu-X)(12)(z-) (X = Br, I; M = Rh, Ir, z = 3; M = Ru, z = 4) with exclusively 2c-2e bonds and pentagonal bipyramidal RhBi7Br8 with a 5c-4e bond in the equatorial pentagonal plane indicative of Mobius aromaticity. The compound Ru3Bi24Br20 contains a more complicated discrete bismuth cluster ion Ru2Bi17(mu-Br)(4)(5+), which can be dissected into a RuBi5 closo octahedron and a RuBi8 nido capped square antiprism bridged by a Ru2Bi4(mu-Br)(4) structural unit. In RuBi4X2 (X = Br, 1), the same Ru2Bi4(mu-Br)(4) structural unit bridges Bi-4 squares similar to those found in the known Zintl ion Bi-4(2-) to give infinite chains of Ru2Bi4 octahedra. The electron counts of the RuBi5, RuBi8, and Ru2Bi4 polyhedra in these structures follow the Wade-Mingos rules. A different infinite chain structure is constructed from fused RhBi7/2Bi bicapped trigonal prisms in Rh2Bi9Br3. This Rh2Bi9Br3 structure can alternatively be derived from alternating Rh2/2Bi4 octahedra and Rh2/2Bi5 pentagonal bipyramids with electron counts obeying the Wade-Mingos rules. Related chemical bonding principles appear to apply to more complicated layer structures such as Pt3Bi13I7 containing Kagome nets of PtBi8/2 cubes and Ni4Bi12X3 containing linked chains of NiBi6/3Bi capped trigonal prisms.