Semiconductor colloidal quantum dots (CQDs) have attracted vast scientific and technological interest throughout the past three decades, due to the unique tuneability of their optoelectronic properties by variation of size and composition. However, the nanoscale size brings about a large surface-to-bulk volume ratio, where exterior surfaces have a pronounced influence on the chemical stability and on the physical properties of the semiconductor. Therefore, numerous approaches have been developed to gain efficient surface passivation, including a coverage by organic or inorganic molecular surfactants as well as the formation of core/shell heterostructures (a semiconductor core epitaxially covered by another semiconductor shell). This review focuses on special designs of core/shell heterostructures from the IV-VI and II-VI semiconductor compounds, and on synthetic approaches and characterization of the optical properties. Experimental observations revealed the formation of core/shell structures with type-I or quasi-type-II band alignment between the core and shell constituents. Theoretical calculations of the electronic band structures, which were also confirmed by experimental work, exposed surplus electronic tuning (beyond the radial diameter) with adaptation of the composition and control of the interface properties. The studies also considered strain effects that are created between two different semiconductors. It was disclosed experimentally and theoretically that the strain can be released via the formation of alloys at the core-shell interface. Overall, the core/shell and core/alloyed-shell heterostructures showed enhancement in luminescence quantum efficiency with respect to that of pure cores, extended lifetime, uniformity in size and in many cases good chemical sustainability under ambient conditions.