A study of CdTe layers grown by molecular beam epitaxy on InSb(111)A (In terminated), and (111)B (Sb terminated) substrates is reported. The growth of CdTe on InSb(111) substrates was studied in situ by reflection high-energy electron diffraction, and characterized by atomic force microscopy, Raman spectroscopy, x-ray diffraction, and photoreflectance. We observed a markedly different CdTe growth behavior on InSb(111)A and B surfaces. CdTe grows nearly two dimensionally on the (111)B surface, whereas on the A surface a three-dimensional growth is obtained, resulting in polycrystalline regions with zinc-blende and wurtzite phases. Our results indicate that In-Te compounds are formed at the CdTe/InSb interface. The amount of these compounds is larger on the (111)A face, thus hindering the smooth CdTe growth on this face.