Averievite-type compounds with the general formula (MX)[Cu5O2(TO4)], where M = alkali metal, X = halogen and T = P, V, have been synthesized by crystallization from gases and structurally characterized for six different compositions: 1 (M = Cs; X = Cl; T = P), 2 (M = Cs; X = Cl; T = V), 3 (M = Rb; X = Cl; T = P), 4 (M = K; X = Br; T = P), 5 (M = K; X = Cl; T = P) and 6 (M = Cu; X = Cl; T = V). The crystal structures of the compounds are based upon the same structural unit, the layer consisting of a kagome lattice of Cu2+ ions and are composed from corner-sharing (OCu4) anion-centered tetrahedra. Each tetrahedron shares common corners with three neighboring tetrahedra, forming hexagonal rings, linked into the two-dimensional [O2Cu5]6+ sheets parallel to (001). The layers are interlinked by (T5+O4) tetrahedra (T5+ = V, P) attached to the bases of the oxocentered tetrahedra in a "face-to-face" manner. The resulting electroneutral 3D framework {[O2Cu5](T5+O4)2}0 possesses channels occupied by monovalent metal cations M+ and halide ions X-. The halide ions are located at the centers of the hexagonal rings of the kagome nets, whereas the metal cations are in the interlayer space. There are at least four different structure types of the averievite-type compounds: the P-3m1 archetype, the 2 × 2 × 1 superstructure with the P-3 space group, the monoclinically distorted 1 × 1 × 2 superstructure with the C2/c symmetry and the low-temperature P21/c superstructure with a doubled unit cell relative to the high-temperature archetype. The formation of a particular structure type is controlled by the interplay of the chemical composition and temperature. Changing the chemical composition may lead to modification of the structure type, which opens up the possibility to tune the geometrical parameters of the kagome net of Cu2+ ions.