Quasar absorption spectra provide the best probe of the state of the gaseous universe at high redshifts. Lyman-α absorption lines map out hydrogen gas structures ranging from tenuous filaments (forest lines) to galaxies (damped absorbers). These are naturally occurring features in hierarchical structure formation theories and well described by the cosmic web picture. A fast non-periodic hydrodynamical N-body code, Tree-P 3M-SPH, was developed and extensively tested. Simulations were performed for the standard cold dark matter cosmological model and the hot-cold, open and vacuum extensions. Artificial spectra were generated and fit as observed spectra are, with an automated Voigt profile line fitter, allowing a direct confrontation of theory with data. Both high resolution and the inclusion of large-scale effects are shown to be essential. These competing aims are satisfied here with many simulations of small patches of the universe and the accurate inclusion of long waves and tides. Mean shear is identified as the critical parameter describing the patches. The results are combined into a representative sample using the theory of Gaussian random fields. The advantages of constrained field initial conditions for high resolution structure formation studies are demonstrated. The cosmological models studied differ in power spectrum shape, amplitude and dark matter abundances. After rescaling the ultraviolet flux, each model fits the observed flux depression and line column density distribution in detail. Non-local measures, such as two point functions, would better reflect the highly visible differences between the models. The range of ultra-violet fluxes required falls within the broad observational constraints. Changing the ultra-violet flux history is found to have a measurable impact on the gas temperatures only at high densities. The mini-halo picture of absorbers is confirmed; at redshift z = 3, the dominant absorbers are dwarf galaxies for neutral hydrogen columns $NHI\sim 1015-1017cm-2$ and filamentary gas below $NHI\sim 1015cm-2.$ Insufficient numerical resolution is shown to suppress or remove the dwarf galaxies. The time evolution, helium absorption, line width versus column density and line width distributions match the observations well for cosmological models with similar fluctuation amplitudes on small scales.