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 spectra , their coherent long-range energy transport, their conceptual similarity to natural light-harvesting complexes , 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 superlattices . Caesium lead halide perovskite nanocrystals are promising building blocks for such superlattices, owing to the high oscillator strength of bright triplet excitons , slow dephasing (coherence times of up to 80 picoseconds) and minimal inhomogeneous broadening of emission lines . So far, only single-component superlattices with simple cubic packing have been devised from these nanocrystals . Here we present perovskite-type (ABO ) binary and ternary nanocrystal superlattices, created via the shape-directed co-assembly of steric-stabilized, highly luminescent cubic CsPbBr nanocrystals (which occupy the B and/or O lattice sites), spherical Fe O or NaGdF nanocrystals (A sites) and truncated-cuboid PbS nanocrystals (B sites). These ABO superlattices, as well as the binary NaCl and AlB superlattice structures that we demonstrate, exhibit a high degree of orientational ordering of the CsPbBr 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.