Recent developments in the synthesis of colloidal semiconductor nanocrystals (NCs), also known as quantum dots (QDs), have led to an excellent control over their size and shape, often with a (nearly) atomic precision.[1-3] This level of control turned out to be the key to their practical applications in technologies that require precise control over the properties of emitted light, such as its energy, color purity, polarization, directionality, etc.[4,5] In contrast to the great experimental achievements in the colloidal synthesis and characterization techniques, little remains known about the detailed atomistic structure of colloidal QDs, especially the structure of the nanocrystal-ligand interface, which is paramount for controlling their size and shape, optical properties and environmental stability. Focusing on CsPbBr₃ NCs as a representative of a recently discovered family of ionic lead halide perovskite QDs, we will present the first computational investigation of the equilibrium structures of realistically sized (≈4 nm) QDs using large-scale classical force-field molecular dynamics (MD) simulations in explicit solvent. The NCs are predicted to have an inherently CsBr-rich composition and equilibrium nearly cubic shape with the main facets of the {100}_p type (with respect to the primitive unit cell), whereas the analogous PbBr₂-terminated nanocubes are found to be unstable and phase separating with the formation of PbBr₂-rich material. These results agree with previous experimental observations and the fact that nanocube is the most frequently encountered shape for CsPbBr₃ NCs reported in the literature.[6,7] Exploration of the entire phase diagram in terms of NC composition and size further allowed us to shed light on the influence of composition on the NC shape, revealing the presence and structures of the minor {111}_p and {110}_p facets that cause truncation and chamfering of the nanocubes at low and high contents of CsBr. The structures and relative occurrence of different crystallographic facets are rationalized using the concept of nonpolar and polar crystal surfaces. Finally, a preferential binding of organic ligands to the different crystallographic facets and its effect on the NC shape will be discussed. The generated ensembles of representative NC structures will serve as a basis for further investigations of structure-property relationships in these nanomaterials, in particular the influence of surface chemistry and structural defects on the optical properties of the NCs.