We introduce stoichiometry control within both core and shell regions of InP/ZnSe/ZnS core/shell/shell quantum dots (QDs) to advance their properties drastically, approaching those of state-of-the-art CdSe-based QDs. The resulting QDs possess near-unity photoluminescence quantum yield, monoexponential decay dynamics, narrow line width, and nonblinking at a single-dot level. Quantum-dot light-emitting diodes (QLEDs) with the InP/ZnSe/ZnS core/shell/shell QDs as emitters exhibit a peak external quantum efficiency of 12.2% and a maximum brightness of >10 000 cd m<sup>-2</sup>, greatly exceeding those of the Cd/Pb-free QLEDs reported in literature. These results pave the way toward Cd/Pb-free QDs as outstanding optical and optoelectronic materials.