In this paper, we review recent progress on colloidal growth of 2D nanocrystals. We identify the four main sources of anisotropy which lead to the formation of plate- and sheet-like colloidal nanomaterials. Defect-induced anisotropy is a growth method which relies on the presence of topological defects at the nanoscale to induce 2D shapes objects. Such a method is particularly important in the growth of metallic nano-objects. Another way to induce anisotropy is based on ligand engineering. The availability of some nanocrystal facets can be tuned by selectively covering the surface with ligands of tunable thickness. Cadmium chalcogenides nanoplatelets (NPLs) strongly rely on this method which offers atomic control in the thinner direction, down to a few monolayers. Two-dimensional objects can also be obtained by post or in situ self-assembly of nanocrystals. This growth method differs from the previous ones in the sense that the elementary objects are not molecular precursors and is a common method for lead chalcogenide compounds. Finally, anisotropy may simply rely on the lattice anisotropy itself as it is common for rod-like nanocrystals. Colloidally grown transition metal dichalcogenides (TMDC) in particular result from such process. We also present hybrid syntheses which combine several of the previously described methods and other paths, such as cation exchange, which expand the range of available materials. Finally, we discuss in which sense 2D objects differ from 0D nanocrystals and review some of their applications in optoelectronics, including lasing and photodetection, and biology.