Core/shell semiconductor nanocrystals with InAs cores were synthesized and characterized. III-V semiconductor shells (InP and GaAs), and II - VI semiconductor shells (CdSe, ZnSe, and ZnS) were overgrown on InAs cores with various radii using a two step synthesis. In the first step cores were prepared, and in the second step the shells were grown using high-temperature pyrolysis of organometallic precursors in a coordinating solvent. Core/shell growth was monitored by absorption and photoluminescence spectroscopy. The band gap shifts to the red upon growth of InP or CdSe shells, while for ZnSe and ZnS shells that have larger band offsets with respect to InAs, the band gap energy is maintained. This behavior is reproduced by band gap energy calculations using a particle within a spherical box model. The photoluminescence quantum yield is quenched in InAs/InP core/shells but increases substantially up to 20% for InAs/CdSe and InAs/ZnSe core/shells. For InAs/ZnS core/shells the enhancement of the photoluminescence quantum yields is smaller, up to 8%. The core/shell nanocrystals were characterized using transmission electron microscopy, X-ray photoelectron spectroscopy, and powder X-ray diffraction. X-ray photoelectron spectroscopy provides evidence for shell growth. The X-ray diffraction peaks shift and narrow upon shell growth, providing evidence for an epitaxial growth mode. Simulations of the X-ray diffraction patterns reproduce both effects, and show that there is one stacking fault present for every four to five layers in the core and core/shell nanocrystals. The stability of InAs/CdSe and InAs/ZnSe core/shells against oxidation is substantially improved compared with the cores, and the photostability is significantly better compared with a typical near-IR laser dye IR140. Core/shell nanocrystals with InAs cores are suggested as a novel type of fluorophores covering the near-IR spectral range, with high emission quantum yields and improved stability compared with traditional near-IR laser dyes.