The majority of stars in the universe has formed in disc galaxies with masses similar to that of the Milky Way. Ab-initio cosmological hydrodynamical simulations of the formation and evolution of galaxies in a Lambda Cold Dark Matter universe have long suffered from serious problems in correctly modelling the star-formation history and structure of disc galaxies. We first use idealized semi-cosmological simulations to gain a better understanding of processes leading to problems in disc formation simulations. We add rotating spheres of hot gas to cosmological dark-matter-only simulations of individual haloes and follow the formation and evolution of galaxy discs from the cooling gas. The initial orientation of the baryonic angular momentum with respect to the halo has a major effect on disc formation. Despite the coherently rotating initial conditions, the orientations of the disc and the outer gas and the relative angle between the components can all change by more than 90 degrees over several billion years. Dominant discs with realistic structural and kinematical properties form preferentially if slow cooling times shift disc formation to later times, if the initial angular momentum is aligned with the halo minor axis and if there is little reorientation of the disc. We then present a new set of fully cosmological simulations with an updated multiphase smoothed particle hydrodynamics galaxy formation code. The update includes improved treatment of metal-line cooling, metal production, turbulent diffusion of metals, kinetic and thermal supernova feedback and radiation pressure from massive young stars. We compare the models to a variety of observations at high and low redshifts and find good agreement for morphologies, stellar-to-dark-matter mass ratios, star formation rates, gas fractions and heavy element abundances. Agreement is better at redshift z=1 than at present day as discrepancies in star formation histories for the lowest and highest simulated galaxy masses become apparent at late times. 18 out of 19 of our model galaxies at z=0 contain stellar discs with kinematic disc fractions up to 65 %, higher than in any previous simulations. We finally compare our model galaxies in detail with recent observations of the structural evolution of stellar galactic discs and the structure of z=0 gas discs. Stellar surface density profiles agree well with observations at z>1, but reveal too little central growth afterwards. This is likely connected to a lack of bars in our simulations resulting from overly strong feedback. Discs at z=0 are too extended by a factor \sim 2. The discs have diverse formation histories ranging from pure inside-out growth in systems with quiescent merger histories to continuous mass growth at all radii. Central mass growth in our models is driven by mergers and misaligned infall events, which leave signatures in the present day distributions of radii and element abundances as functions of stellar age. Gas discs agree well with observations in terms of sizes and profile shapes, but on average have overly high gas-to-stellar mass ratios. Our models agree well with the observed neutral hydrogen mass-size relation. Despite significant progress, our models continue to suffer from various problems illustrating that we are still far away from capturing all relevant physical processes accurately.