Abstract. Using high resolution (R ∼ 45 000), high S/N (∼20-50) VLT/UVES data, we have analyzed the Lyα forest of 3 QSOs in the neutral hydrogen (H i) column density range NH i = 10 12.5−16 cm −2 at 1.5 < z < 2.4. We combined our results with similar high-resolution, high S/N data in the literature at z > 2.4 to study the redshift evolution of the Lyα forest at 1.5 < z < 4. We have applied two types of analysis: the traditional Voigt profile fitting and statistics on the transmitted flux. The results from both analyses are in good agreement: 1. The differential column density distribution function, f (NH i), of the Lyα forest shows little evolution in the column density range NH i = 10 12.5−14 cm −2 , f (NH i) ∝ N −β H i , with β ∼ 1.4-1.5 at 1.5 < z < 4 and with a possible increase of β to β ∼ 1.7 at z < 1.8. A flattening of the power law slope at lower column densities at higher z can be attributed to more severe line blending. A deficiency of lines with NH i > 10 14 cm −2 is more noticeable at lower z than at higher z. The one-point function and the two-point function of the flux confirm that strong lines do evolve faster than weak lines; 2. The line number density per unit redshift, dn/dz, at NH i = 10 13.64−16 cm −2 is well fitted by a single power law, dn/dz ∝ (1 + z) 2.19±0.27 , at 1.5 < z < 4. In combination with the HST results from the HST QSO absorption line key project, the present data indicate that a flattening in the number density evolution occurs at z ∼ 1.2. The line counts as a function of the filling factor at the transmitted flux F in the range 0 < F < 0.9 are constant in the interval 1.5 < z < 4. This suggests that the Hubble expansion is the main drive governing the forest evolution at z > 1.5 and that the metagalactic UV background changes more slowly than a QSO-dominated background at z < 2; 3. The observed cutoff Doppler parameter at the fixed column density NH i = 10 13.5 cm −2 , bc,13.5, shows a weak increase with decreasing z, with a possible local bc,13.5 maximum at z ∼ 2.9; 4. The two-point velocity correlation function and the step optical depth correlation function show that the clustering strength increases as z decreases; 5. The evolution of the mean H i opacity, τ H i, is well approximated by an empirical power law, τ H i ∝ (1+z) 3.34±0.17 , at 1.5 < z < 4; 6. The baryon density, Ω b , derived both from the mean H i opacity and from the one-point function of the flux is consistent with the hypothesis that most baryons (over 90%) reside in the forest at 1.5 < z < 4, with little change in the contribution to the density, Ω, as a function of z.