This thesis deals with the growth of GaAs nanowires (NWs) by molecular beam epitaxy (MBE) using vapor-liquid-solid method on various substrates including GaAs(111)B, Si(111) and graphene. The growth of the NWs on GaAs substrates was carried out by Au-catalyzed technique, whereas the growths on Si and graphene substrates were carried out using self-catalyzed technique that has been the main focus of this thesis. The long-term goal of this work was to produce p-n radial junction GaAs NWs for solar cell applications.Necessary conditions were established for obtaining vertical self-catalyzed GaAs NWs on Si(111), which is reproducible from run-to-run. One of the major issues in these NWs grown by both Au-and self-catalyzed techniques is their crystal structure. The Au-catalyzed GaAs NWs usually adopt a wurtzite (WZ) crystal phase, whereas the self-catalyzed NWs a zinc blende (ZB) phase. However, in both the cases the NWs contain stacking faults, rotational twins or/and a mixed crystal phase. The ZB and WZ phases show different optical properties, and one phase might be favored over other for certain applications. Therefore the crystal phase was controlled within single NWs by tuning the V/III ratio and introducing GaAsSb inserts. The change of the crystal phases was correlated with the change in the contact angle of the Ga droplet.Since the discovery of graphene, an ultra-thin two-dimensional material, the research on graphene has become an active field in recent years due to its remarkable properties including excellent electrical and thermal conductivities, mechanical strength and flexibility, and optical transparency. By growing the semiconductor NWs on graphene, a completely new hybrid system can be envisioned where the unique properties of both NWs and graphene can be utilized. Therefore we established a method for the growth of semiconductor NWs on graphene by demonstrating epitaxial growth of vertical GaAs and InAs NWs on different graphitic substrates.Core-shell heterostructure, doping, optical properties, and position controlled growth of self-catalyzed GaAs NWs were investigated. Growth of GaAs/GaAsSb coreshell NWs where the Sb content was tuned from about 10% -70% was studied. The effect of growth temperature and the Sb flux on the morphology of GaAsSb shell was investigated. In addition, by utilizing the core-shell geometry where the shell copies the crystal phase of the core, WZ phase of GaAsSb was demonstrated. Successful p-type doping of GaAs core using Be as dopant, and n-type doping of GaAs shell using Si and Te as dopants were achieved. To investigate the optical properties, GaAs/AlGaAs coreshell NWs were grown with different V/III ratios during the core growth. The NWs grown with high V/III ratio, despite containing a higher density of twinned ZB and WZ GaAs with SFs, were found to have superior optical quality as compared to the NWs grown with low V/III ratio that contain pure ZB GaAs. The observed V/III ratio dependent optical quality was correlated to the intrinsic defects such as As vac...