Colloidal lead halide perovskite nanocrystals (LHP NCs, $APbX_3$, where $A=Cs^+$, $FA^+$, FA=formamidinium; X=Cl, Br, I) have become a research spotlight owing to their spectrally narrow (~100 meV) fluorescence, tunable over the entire visible spectral region of 400-800 nm, as well as facile colloidal synthesis. These NCs are attractive single-photon emitters and building blocks for creating controlled, aggregated states exhibiting collective luminescence phenomena. Attaining such states through the spontaneous self-assembly into long-range ordered superlattices (SLs) is a particularly attractive avenue. The atomically flat, sharp cubic shape of LHP NCs is also of interest because the vast majority of prior work had invoked NCs of spherical shape. Long-range ordered SLs with the simple cubic packing of cubic perovskite NCs exhibit sharp red-shifted lines in their emission spectra and superfluorescence (a fast collective emission resulting from coherent multi-NCs excited states). When $CsPbBr_3$ NCs are combined with spherical dielectric NCs, perovskite-type $ABO_3$ binary NC SLs form, wherein $CsPbBr_3$ nanocubes occupy B- and/or O-sites, while spherical dielectric $Fe_3O_4$ or $NaGdF_4$ NCs reside on A-sites. When truncated- cuboid PbS NCs are added to these systems, ternary $ABO_3$ phase form (PbS NCs occupy B-sites). Such $ABO_3$ SLs, as well as other newly obtained SL structures (binary NaCl, $AlB_2$- and $ABO_6$ types, columnar assemblies with disks, etc.), exhibit a high degree of orientational ordering of $CsPbBr_3$ nanocubes. These mesostructures also exhibit superfluorescence, characterized, at high excitation density, by emission pulses with ultrafast (22 ps) radiative decay and Burnham- Chiao ringing behavior with a strongly accelerated build-up time. Co-assembly of steric-stabilized $CsPbBr_3$ nanocubes with disk-shaped $LaF_3$ NCs yields six columnar structures with AB, $AB_2$, $AB_4$, and $AB_6$ stoichiometry, not observed before with NC systems comprising spheres and disks. Combining $CsPbBr_3$ nanocubes with large and thick $NaGdF_4$ nanodisks results in the orthorhombic SL resembling $CaC_2$ structure with pairs of $CsPbBr_3$ NCs on one lattice site. Additionally, we have also implemented two substrate-free methods of SL formation. The first method involves oil-in-oil templated assembly that leads to the formation of binary supraparticles, while the second method utilizes self- assembly at the liquid-air interface.