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This thesis focuses on the synthesis, structures, and physical properties of ternary rare-earth antimonides and germanides. These ternary compounds exhibit diverse polyanionic substructures with classical and non-classical Sb–Sb bonding, Ge–Ge bonding, or both. The Zintl-Klemm concept and band structure calculations were applied to understand their structures and bonding. Electrical resistivities and magnetic properties were measured for these compounds. The compounds RE2Ti7Sb12 (RE = La–Nd) and RE2Ti11–xSb14+x (RE = Sm, Gd, Tb, Yb), which were synthesized by arc-melting, adopt different structures depending on the size of the RE atoms. Both consist of a complex arrangement of TiSbn polyhedra, linked to form a 3D framework with large cavities in which the RE atoms reside. Hypervalent Sb–Sb bonds are manifested in disordered Sb fragments in RE2Ti7Sb12, and 1D linear chains, zig-zag chains, and pairs in RE2Ti11–xSb14+x. A series of compounds, RECrGe3 (RE = La–Nd, Sm), was synthesized by the Sn-flux method. They adopt a hexagonal perovskite structure type, in which chains of face-sharing Cr-centred octahedra are linked by triangular Ge3 clusters. These unusual single-bonded Ge3 substructures can be rationalized simply by the Zintl-Klemm concept. Electrical resistivity measurements show metallic behaviour with prominent transitions coincident with ferromagnetic transitions (Tc ranging from 62 to 155 K) found in magnetic measurements. Band structure calculations show the presence of a narrow, partially filled band with high DOS at Ef, in agreement with the observation that LaCrSb3 is an itinerant ferromagnet. With a different number of d-electrons in the M site, the isostructural REVGe3 compounds exhibit antiferromagnetic behaviour. The doped quaternary compounds LaCr1-xVxGe3 and LaCr1-xMnxGe3 exhibit depressed Curie temperatures. The structures of RECrxGe2 compounds (RE = Sm, Gd–Er) are built up by inserting transition-metal atoms into the square pyramidal sites of a hypothetical “REGe2” host structure (ZrSi2-type). The presence of extensive anionic Ge substructures in the form of 1D zigzag chains and 2D square sheets can be explained by the Zintl-Klemm concept. Magnetic measurements indicated antiferromagnetic ordering with low TN ranging from 3 to 17 K. Compounds involving a p-block element as the second component were prepared. In RE12Ge7–xSb21 (RE = La–Pr), a complex 3D polyanionic framework with Ge pairs, five-atom-wide Sb ribbons, and 2D Ge/Sb layers is present. The bonding exemplifies the competition of valence electron transfer from the RE atoms to metalloids with similar electronegativities. Full electron transfer from the RE atoms to the anionic substructure cannot be assumed. Magnetic measurements on Ce12Ge6.5Sb21 indicate antiferromagnetic coupling. The metal-rich compounds RE5TtxSb3–x (Tt = Si, Ge) adopt the orthorhombic beta-Yb5Sb3-type structure with a range of solid solubility between 0.9 and 1.6. They are not electron-precise and do not obey the Zintl-Klemm concept.