We present data and initial results from VLT/X-Shooter emission-line spectroscopy of 96 galaxies selected by long γ-ray bursts (GRBs) at 0.1 <z< 3.6, the largest sample of GRB host spectra available to date. Most of our GRBs were detected by Swift and 76% are at 0.5 <z< 2.5 with a median zmed ~ 1.6. Based on Balmer and/or forbidden lines of oxygen, nitrogen, and neon, we measure systemic redshifts, star formation rates (SFR), visual attenuations (AV), oxygen abundances (12 + log (O/H)), and emission-line widths (σ). We study GRB hosts up to z ~ 3.5 and find a strong change in their typical physical properties with redshift. The median SFR of our GRB hosts increases from SFRmed ~ 0.6 M⊙ yr-1 at z ~ 0.6 up to SFRmed ~ 15 M⊙ yr-1 at z ~ 2. A higher ratio of O iii/O ii  at higher redshifts leads to an increasing distance of GRB-selected galaxies to the locus of local galaxies in the Baldwin-Phillips-Terlevich diagram. There is weak evidence for a redshift evolution in AV and σ, with the highest values seen at z ~ 1.5 (AV) or z ~ 2 (σ). Oxygen abundances of the galaxies are distributed between 12 + log (O/H) = 7.9 and 12 + log (O/H) = 9.0 with a median 12 + log (O/H)med ~ 8.5. The fraction of GRB-selected galaxies with super-solar metallicities is ~20% at z< 1 in the adopted metallicity scale. This is significantly less than the fraction of total 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⊙. Together with a high incidence of Z ~ 0.5 Z⊙ galaxies at z ~ 1.5, this indicates that a metallicity dependence at low redshift will not be dominant at z ~ 3. Significant correlations exist between the hosts’ physical properties. Oxygen abundance, for example, relates to AV (12 + log (O/H) ∝ 0.17·AV), line width (12 + log (O/H) ∝ σ0.6), and SFR (12 + log (O/H) ∝ SFR0.2). In the last two cases, the normalization of the relations shift to lower metallicities at z> 2 by ~0.4 dex. These properties of GRB hosts and their evolution with redshift can be understood in a cosmological context of star-forming galaxies and a picture in which the hosts’ properties at low redshift are influenced by the tendency of GRBs to avoid the most metal-rich environments.