Here we report on the Ga self-catalyzed growth of near full-composition-range energy-gap-tunable GaAsSb nanowires by molecular-beam epitaxy. GaAsSb nanowires with different Sb content are systematically grown by tuning the Sb and As fluxes, and the As background. We find that GaAsSb nanowires with low Sb content can be grown directly on Si(111) substrates (0 ≤ x ≤ 0.60) and GaAs nanowire stems (0 ≤ x ≤ 0.50) by tuning the Sb and As fluxes. To obtain GaAsSb nanowires with x ranging from 0.60 to 0.93, we grow the GaAsSb nanowires on GaAs nanowire stems by tuning the As background. Photoluminescence measurements confirm that the emission wavelength of the GaAsSb nanowires is tunable from 844 nm (GaAs) to 1760 nm (GaAsSb). High-resolution transmission electron microscopy images show that the grown GaAsSb nanowires have pure zinc-blende crystal structure. Room-temperature Raman spectra reveal a redshift of the optical phonons in the GaAsSb nanowires with x increasing from 0 to 0.93. Field-effect transistors based on individual GaAsSb nanowires are fabricated, and rectifying behavior is observed in devices with low Sb content, which disappears in devices with high Sb content. The successful growth of high-quality GaAsSb nanowires with near full-range bandgap tuning may speed up the development of high-performance nanowire devices based on such ternaries.