The II-VI compound semiconductor CdSe has a bandgap energy of 1.71 eV  and 1.68 eV  in the wurtzite (hexagonal) and zincblende (cubic) crystal structures, respectively, making it an ideal candidate material for the top cell in tandem application with a Si bottom cell. However, the growth of monocrystalline CdSe and control of its phase, wurtzite or zincblende, remains a challenge. Molecular beam epitaxy (MBE) growth of CdSe thin films nearly lattice matched to InAs (100), (111)A, and (111)B oriented substrates is investigated. Growth temperature ranges from 250 to 350 °C, Cd/Se flux ratio ranges from 0.74 to 1.35, and the growth rate ranges from 0.14 to 0.84 monolayers per second. Single crystal zincblende material luminescing at 1.668 eV is demonstrated on the (100) substrates, while polycrystalline mixed-phase material luminescing from 1.589 to 1.726 eV is demonstrated on the (111) substrates. In-situ reflection high energy electron diffraction (RHEED) patterns show a clear transition from zincblende 1 times 1 surface reconstructions with four-fold symmetry to wurtzite 1 times 1 reconstructions with six-fold symmetry. The results indicate that CdSe crystal phase and thin film morphology is highly sensitive to growth temperature, Cd/Se flux ratio, and polar (111) surface preparation.