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 z med ∼ 1.6. Based on Balmer and/or forbidden lines of oxygen, nitrogen, and neon, we measure systemic redshifts, star formation rates (SFR), visual attenuations (A V ), 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 SFR med ∼ 0.6 M yr −1 atO ] 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 A V and σ, with the highest values seen at z ∼ 1.5 (A V ) 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 A V (12 + log(O/H) ∝ 0.17 · A V ), line width (12 + log(O/H) ∝ σ 0.6 ), and SFR (12 + log(O/H) ∝ SFR 0.2 ). 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.