The K-Au-Ga system has been investigated at 350 degrees C for <50 at. % K. The potassium gold gallides K0.55Au2Ga2, KAu3Ga2, KAu2Ga4 and the solid solution KAuxGa3-x (x = 0-0.33) were synthesized directly from the elements via typical high-temperature reactions, and their crystal structures were determined by single crystal X-ray diffraction: K0.55Au2Ga2 (I, I4/mcm, a = 8.860(3) angstrom, c = 4.834(2) angstrom, Z = 4), KAu3Ga2 (II, Cmcm, a = 11.078(2) angstrom, b = 8.486(2) angstrom, c = 5.569(1) angstrom, Z = 4), KAu2Ga4 (III, Immm, a = 4.4070(9) angstrom, b = 7.339(1) angstrom, c = 8.664(2) angstrom, Z = 2), KAu0.33Ga2.67 (IV, I-4m2, a = 6.0900(9) angstrom, c = 15.450(3) angstrom, z = 6). The first two compounds contain different kinds of tunnels built of puckered six- (II) or eight-membered (I) ordered Au/Ga rings with completely different cation placements: uniaxial in I and III but in novel 2D-zigzag chains in H. III contains only infinite chains of a potassium-centered 20-vertex polyhedron (K@Au8Ga12) built of ordered 6-8-6 planar Au/Ga rings. The main structural feature of IV is dodecahedral (Au/Ga)(8) clusters. Tight-binding electronic structure calculations by linear muffin-tin-orbital methods were performed for idealized models of I, II, and III to gain insights into their structure-bonding relationships. Density of states curves reveal metallic character for all compounds, and the overall crystal orbital Hamilton populations are dominated by polar covalent Au-Ga bonds. The relativistic effects of gold lead to formation of bonds of greater population with most post-transition elements or to itself, and these appear to be responsible for a variety of compounds, as in the K-Au-Ga system.