The field of colloidal synthesis of semiconductors emerged 40 years ago and has reached a certain level of maturity thanks to the use of nanocrystals as phosphors in commercial displays. In particular, II-VI semiconductors based on cadmium, zinc, or mercury chalcogenides can now be synthesized with tailored shapes, composition by alloying, and even as nanocrystal heterostructures. Fifteen years ago, II-VI semiconductor nanoplatelets injected new ideas into this field. Indeed, despite the emergence of other promising semiconductors such as halide perovskites or 2D transition metal dichalcogenides, colloidal II-VI semiconductor nanoplatelets remain among the narrowest room-temperature emitters that can be synthesized over a wide spectral range, and they exhibit good material stability over time. Such nanoplatelets are scientifically and technologically interesting because they exhibit optical features and production advantages at the intersection of those expected from colloidal quantum dots and epitaxial quantum wells. In organic solvents, gram-scale syntheses can produce nanoparticles with the same thicknesses and optical properties without inhomogeneous broadening. In such nanoplatelets, quantum confinement is limited to one dimension, defined at the atomic scale, which allows them to be treated as quantum wells. In this review, we discuss the synthetic developments, spectroscopic properties, and applications of such nanoplatelets. Covering growth mechanisms, we explain how a thorough understanding of nanoplatelet growth has enabled the development of nanoplatelets and heterostructured nanoplatelets with multiple emission colors, spatially localized excitations, narrow emission, and high quantum yields over a wide spectral range. Moreover, nanoplatelets, with their large lateral extension and their thin short axis and low dielectric surroundings, can support one or several electron-hole pairs with large exciton binding energies. Thus, we also discuss how the relaxation processes and lifetime of the carriers and excitons are modified in nanoplatelets compared to both spherical quantum dots and epitaxial quantum wells. Finally, we explore how nanoplatelets, with their strong and narrow emission, can be considered as ideal candidates for pure-color light emitting diodes (LEDs), strong gain media for lasers, or for use in luminescent light concentrators.