AbstractAbsorption‐line systems detected in high resolution quasar spectra can be used to compare the value of dimensionless fundamental constants such as the fine‐structure constant, α, and the proton‐to‐electron mass ratio, μ = mp/me, as measured in remote regions of the Universe to their value today on Earth. In recent years, some evidence has emerged of small temporal and also spatial variations in α on cosmological scales which may reach a fractional level of ≈ 10 ppm (parts per million). We are conducting a Large Programme of observations with the Very Large Telescope's Ultraviolet and Visual Echelle Spectrograph (UVES), and are obtaining high‐resolution (R ≈ 60000) and high signal‐to‐noise ratio (S/N ≈ 100) spectra calibrated specifically to study the variations of the fundamental constants. We here provide a general overview of the Large Programme and report on the first results for these two constants, discussed in detail in Molaro et al. (2013) and Rahmani et al. (2013). A stringent bound for Δα /α is obtained for the absorber at zabs = 1.6919 towards HE 2217‐2818. The absorption profile is complex with several very narrow features, and is modeled with 32 velocity components. The relative variation in α in this system is +1.3 ± 2.4stat ± 1.0sys ppm if Al II λ 1670 Å and three FeII transitions are used, and +1.1 ± 2.6stat ppm in a slightly different analysis with only FeII transitions used. This is one of the tightest bounds on α ‐variation from an individual absorber and reveals no evidence for variation in α at the 3‐ppm precision level (1σ confidence). The expectation at this sky position of the recently‐reported dipolar variation of α is (3.2–5.4) ± 1.7 ppm depending on dipole model used and this constraint of Δα /α at face value is not supporting this expectation but not inconsistent with it at the 3σ level. For the proton‐to‐electron mass ratio the analysis of the H2 absorption lines of the zabs ≈ 2.4018 damped Lyα system towards HE 0027–1836 provides Δμ /μ = (–7.6 ± 8.1stat ± 6.3sys) ppm which is also consistent with a null variation. The cross‐correlation analysis between individual exposures taken over three years and comparison with almost simultaneous asteroid observations revealed the presence of a possible wavelength dependent velocity drift as well as of inter‐order distortions which probably dominate the systematic error and are a significant obstacle to achieve more accurate measurements. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)