We have studied the epitaxial growth of an InP shell on various pure InAs core nanowire crystal structures by metal-organic vapor phase epitaxy. The InP shell is grown on wurtzite (WZ), zinc-blende (ZB), and {111}- and {110}-type faceted ZB twin-plane superlattice (TSL) structures by tuning the InP shell growth parameters and controlling the shell thickness. The growth results, particularly on the WZ nanowires, show that homogeneous InP shell growth is promoted at relatively high temperatures (∼500 °C), but that the InAs nanowires decompose under the applied conditions. In order to protect the InAs core nanowires from decomposition, a short protective InP segment is first grown axially at lower temperatures (420-460 °C), before commencing the radial growth at a higher temperature. Further studies revealed that the InP radial growth rate is significantly higher on the ZB and TSL nanowires compared to WZ counterparts, and shows a strong anisotropy in polar directions. As a result, thin shells were obtained during low temperature InP growth on ZB structures, while a higher temperature was used to obtain uniform thick shells. In addition, a schematic growth model is suggested to explain the basic processes occurring during the shell growth on the TSL crystal structures.