  [Abridged] We present data and initial results from VLT/X-Shooter
emission-line spectroscopy of 96 GRB-selected galaxies at 0.1<z<3.6, the
largest sample of GRB host spectroscopy available to date. Most of our GRBs
were detected by Swift and 76% are at 0.5<z<2.5 with a median z~1.6. Based on
Balmer and/or forbidden lines of oxygen, nitrogen, and neon, we measure
systemic redshifts, star formation rates (SFRs), visual attenuations, oxygen
abundances (12+log(O/H)), and emission-line widths. We find a strong change of
the typical physical properties of GRB hosts with redshift. The median SFR, for
example, increases from ~0.6 M_sun/yr at z~0.6 up to ~15 M_sun/yr at z~2. A
higher ratio of [OIII]/[OII] at higher redshifts leads to an increasing
distance of GRB-selected galaxies to the locus of local galaxies in the BPT
diagram. Oxygen abundances of the galaxies are distributed between
12+log(O/H)=7.9 and 12+log(O/H)=9.0 with a median of 12+log(O/H)~8.5. The
fraction of GRB-selected galaxies with super-solar metallicities is around 20%
at z<1 in the adopted metallicity scale. This is significantly less than the
fraction of star formation in similar galaxies, illustrating that GRBs are
scarce in high-metallicity environments. At z~3, sensitivity limits us to
probing only the most luminous GRB hosts for which we derive metallicities of Z
~< 0.5 Z_sun. Together with a high incidence of galaxies with similar
metallicity in our sample at z~1.5, this indicates that the metallicity
dependence observed at low redshift will not be dominant at z~3.
