We report on low-resolution multi-object spectroscopy of 30 faint targets ( R \simeq 24 - 25 ) in the HDF-S and AXAF deep field obtained with the VLT Focal Reducer/low dispersion Spectrograph ( FORS1 ) . Eight high-redshift galaxies with 2.75 < z < 4 have been identified . The spectroscopic redshifts are in good agreement with the photometric ones with a dispersion \sigma _ { z } = 0.07 at z < 2 and \sigma _ { z } = 0.16 at z > 2 . The inferred star formation rates of the individual objects are moderate , ranging from a few to a few tens M _ { \odot } ~ { } { yr } ^ { -1 } . Five out of the eight high-z objects do not show prominent emission lines . One object has a spectrum typical of an AGN . In the AXAF field two relatively close pairs of galaxies have been identified , with separations of 8.7 and 3.1 proper Mpc and mean redshifts of 3.11 and 3.93 , respectively .