|We will present results from our on-going high-resolution spectroscopic survey of bright GRB afterglows with VLT/UVES. These observations are typically performed following a Swift satellite GRB alert with the VLT Rapid-Response Mode (RRM), which allows for automatic observations of any target with a delay time of 5-10 minutes. The sample of decent signal-to-noise UVES GRB afterglow spectra has now grown to about 10, and is slowly increasing with roughly 2-3 per year. We will discuss various studies based on these data, ranging from intervening CIV and MgII systems, to high-ionization lines near the GRB redshift, and the detection of absorption-line variability, which we show to be caused by UV photons from the GRB afterglow itself in absorbers at kpc distances away from the GRB. Finally, we briefly mention the prospect of taking these studies to higher redshift with X-shooter.