Atomically defined assemblies of dye molecules (such as H and J aggregates) have been of interest for more than 80 years because of the emergence of collective phenomena in their optical spectra1–3, their coherent long-range energy transport, their conceptual similarity to natural light-harvesting complexes4,5, and their potential use as light sources and in photovoltaics. Another way of creating versatile and controlled aggregates that exhibit collective phenomena involves the organization of colloidal semiconductor nanocrystals into long-range-ordered superlattices6. Caesium lead halide perovskite nanocrystals7–9 are promising building blocks for such superlattices, owing to the high oscillator strength of bright triplet excitons10, slow dephasing (coherence times of up to 80 picoseconds) and minimal inhomogeneous broadening of emission lines11,12. So far, only single-component superlattices with simple cubic packing have been devised from these nanocrystals13. Here we present perovskite-type (ABO3) binary and ternary nanocrystal superlattices, created via the shape-directed co-assembly of steric-stabilized, highly luminescent cubic CsPbBr3 nanocrystals (which occupy the B and/or O lattice sites), spherical Fe3O4 or NaGdF4 nanocrystals (A sites) and truncated-cuboid PbS nanocrystals (B sites). These ABO3 superlattices, as well as the binary NaCl and AlB2 superlattice structures that we demonstrate, exhibit a high degree of orientational ordering of the CsPbBr3 nanocubes. They also exhibit superfluorescence—a collective emission that results in a burst of photons with ultrafast radiative decay (22 picoseconds) that could be tailored for use in ultrabright (quantum) light sources. Our work paves the way for further exploration of complex, ordered and functionally useful perovskite mesostructures.