Hydrogen has been largely proposed as a possible fuel for internal combustion engines. The main advantage of burning hydrogen is the absence of carbon-based tailpipe emissions. Hydrogen's wide flammability also offers the advantage of very lean combustion and higher engine efficiency than conventional carbon-based fuels. In order to avoid abnormal combustion modes like pre-ignition and backfiring, as well as air displacement from hydrogen's large injected volume per cycle, direct injection of hydrogen after intake valve closure is the preferred mixture preparation method for hydrogen engines. The current work focused on computational studies of hydrogen injection and mixture formation for direct-injection spark-ignition engines. Hydrogen conditions at the injector's nozzle exit are typically sonic. Initially the characteristics of under-expanded sonic hydrogen jets were investigated in a quiescent environment using both Reynolds-Averaged Navier-Stokes (RANS) and Large-Eddy Simulation (LES) techniques. Various injection conditions were studied, including a reference case from the literature. Different nozzle geometries were investigated, including a straight nozzle with fixed cross section and a stepped nozzle design. LES captured details of the expansion shocks better than RANS and demonstrated several aspects of hydrogen's injection and mixing. In-cylinder simulations were also performed with a side 6-hole injector using 70 and 100 bar injection pressure. Injection timing was set to just after inlet valve closure with duration of 6 μs and 8 μs, leading to global air-to-fuel equivalence ratios ϕ typically in the region of 0.2-0.4. The engine intake air pressure was set to 1.5 bar absolute to mimic boosted operation. It was observed that hydrogen jet wall impingement was always prominent. Comparison with non-fuelled engine conditions demonstrated the degree of momentum exchange between in-cylinder hydrogen injection and air motion. LES highlighted details of hydrogen's spatial distribution throughout the injection duration and up to ignition timing. Higher peak velocities were predicted by LES, especially on the tumble plane. With the employed injection strategy, the areas closer to the cylinder wall were richer in fuel than the centre of the chamber close to the end of compression.