The “forest” of hydrogen and helium Lyman-alpha absorption lines observed in the spectra of distant quasars encodes information on the physics of structure formation, the nature of the dark matter, and the temperature and ionization state of baryons in the Universe. In this talk I will describe the multiple steps needed to connect flux fluctuations in quasar spectra to physical parameters, present an unprecedented suite of thousands of high-resolution hydrodynamical simulations of structure formation with different gas thermal histories along with different free-streaming lengths of (warm) dark matter particles, and use it to perform a statistical comparison of mock spectra with the observed 1D flux power spectrum and other data.