A complex InAs/CdSe/ZnSe core/shell1/shell2 (CSS) structure is synthesized, where the intermediate CdSe buffer layer decreases strain between the InAs core and the ZnSe outer shell. This structure leads to significantly improved fluorescence quantum yield as compared to previously prepared core/shell structures and enables growth of much thicker shells. The shell growth is done using a layer-by-layer method in which the shell cation and anion precursors are added sequentially allowing for excellent control, and a good size distribution is maintained throughout the entire growth process. The CSS structure is characterized using transmission electron microscopy, as well as by X-ray diffraction and X-ray photoelectron spectroscopy which provide evidence for shell growth. The quantum yield for CSS with small InAs cores reaches over 70% - exceptional photoluminescence intensity for Ill-V semiconductor nanocrystals. In larger InAs cores there is a systematic decrease in the quantum yield, with a yield of similar to 40% for intermediate size cores down to a few percent in large cores. The CSS structures also exhibit very good photostability, vastly improved over those of organically coated cores, and transformation into water environment via ligand exchange is performed without significant decrease of the quantum yield. These new InAs/CdSe/ZnSe CSS nanocrystals are therefore promising near-IR chromophores for biological fluorescence tagging and optoelectronic devices.