Colloidal lead halide perovskite nanocrystals (NCs) self-assembled into long-range ordered superlattices (SLs) with simple cubic packing of perovskite nanocubes have been shown to emit superfluorescent light, wherein coherent coupling among several NCs leads to collective emission, resulting in ultrafast bursts of photons. Co-assembly of perovskite NCs into multicomponent SLs with other building blocks might serve as a versatile platform for controlling their collective properties. We show that the use of cubic NCs instead of spherical ones changes the assembly outcome. CsPbBr₃ nanocubes combined with spherical Fe₃O₄ or NaGdF₄ NCs and truncated cuboid PbS NCs form binary SLs of six structure types, namely, NaCl-, AlB₂-, CuAu- as well as uncommon to all-sphere assemblies novel AB₂-, quasi-ternary ABO₃- and ABO₆-types. In these structures, the nanocubes retain orientational coherence. The co-assembly of CsPbBr₃ nanocubes with larger disk-shaped LaF₃ NCs (1.6 nm in thickness) results in the formation of six columnar structures with AB, AB₂, AB₄ and AB₆ stoichiometry. In the systems with comparable dimensions of nanocubes (8.6 nm) and nanodisks (6.5–12.5 nm), other, non-columnar structures are observed, such as ReO₃-type SL, featuring intimate intermixing and face-to-face alignment of disks and cubes. With large and thick NaGdF₄ nanodisks, an orthorhombic SL resembling CaC₂ structure with clusters of CsPbBr₃ NCs was obtained. We also explore an on-liquid assembly method using glyceryl triacetate as a subphase that allows obtaining free-floating SL films comprising perovskite nanocrystals.