After a brief historical introduction, I will describe the first analytical models of the Lyman-alpha forest in the framework of Cold Dark Matter scenarios. Then, I will focus on possible ways of extracting cosmological parameters from a set of observed quasar spectra, by running a grid of cosmological hydro-dynamical simulations. In particular, I will recover the linear dark matter power spectrum at z>2 and at scales of ~ 1-40 co-moving Mpc. I will address the significance of the results obtained especially when combined with the larger scale measurement of the power spectrum made by WMAP, giving constraints on the power spectrum amplitude, spectral index and its running. Results in terms of inflationary parameters and masses of elementary particles (e.g. neutrinos and gravitinos) will also be presented. I will critically compare all the results obtained with those of the SDSS collaboration, based on a set of more than 3000 quasars at low resolution. Several physical aspects, which affect the constraints on cosmological parameters, will be discussed: feedback effects in the form of galactic winds, metal enrichment, the thermal state of the Intergalactic Medium and the amplitude and nature of the ultraviolet background.