Absorption-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 ≈ 60 000) 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. and Rahmani et al.A stringent bound for ∆α/α is obtained for the absorber at z abs = 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 1670Å and three Fe II transitions are used, and +1.1 ±2.6stat ppm in a slightly different analysis with only Fe II 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 (1-σ 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 ∆α/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 z abs ≈ 2.4018 damped Lyα system towards HE 0027-1836 provides ∆µ/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.