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Long-duration gamma-ray bursts (GRBs) are powerful tracers of star-forming galaxies. We have defined a homogeneous subsample of 69 Swift GRB-selected galaxies spanning a very wide redshift range. Special attention has been devoted to making the sample optically unbiased through simple and well-defined selection criteria based on the high-energy properties of the bursts and their positions on the sky. Thanks to our extensive follow-up observations, this sample has now achieved a comparatively high degree of redshift completeness, and thus provides a legacy sample, useful for statistical studies of GRBs and their host galaxies. In this paper we present the survey design and summarize the results of our observing program conducted at the ESO Very Large Telescope (VLT) aimed at obtaining the most basic properties of galaxies in this sample, including a catalog of R and Ks magnitudes and redshifts. We detect the host galaxies for 80 % of the GRBs in the sample, although only 42 % have Ks-band detections, which confirms that GRB-selected host galaxies are generally blue. The sample is not uniformly blue, however, with two extremely red objects detected. Moreover, galaxies hosting GRBs with no optical/NIR afterglows, whose identification therefore relies on X-ray localizations, are significantly brighter and redder than those with an optical/NIR afterglow. Our spectroscopic campaign has resulted in 77 % now having redshift measurements, with a median redshift of 2.14 +- 0.18. TOUGH alone includes 17 detected z > 2 Swift GRB host galaxies suitable for individual and statistical studies. Seven hosts have detections of the Ly-alpha emission line and we can exclude an early indication that Ly-alpha emission is ubiquitous among GRB hosts, but confirm that Ly-alpha is stronger in GRB-selected galaxies than in flux-limited samples of Lyman break galaxies.