Remarkable progress has been made over the last few decades in furthering our understanding of the growth of cosmic structure. Nonetheless, there remains a great deal of uncertainty regarding the precise details of the complex baryonic physics that regulate galaxy formation. Any theory of star formation in galaxies must encompass the radiative cooling of gas into dark matter haloes, the formation of a turbulent, multiphase interstellar medium (ISM), the efficiency with which molecular gas is able to collapse into cores and ultimately stars, and the subsequent interaction of those stars with the gas through ionizing radiation, winds and supernova (SN) explosions. Given the highly non-linear nature of the problem, numerical simulations provide an invaluable tool with which to study galaxy formation. Yet, even with contemporary computational resources, the inherently large dynamical range of spatial scales that must be tackled makes the development of such models extremely challenging, inevitably leading to the adoption of `subgrid' approximations at some scale. In this thesis, I explore new methods of incorporating the physics of star formation and stellar feedback into high resolution hydrodynamic simulations of galaxies. I first describe a new implementation of star formation and SN feedback that I have developed for the state-of-the-art moving mesh code Arepo. I carry out a detailed study into various classes of subgrid SN feedback schemes commonly adopted in the literature, including injections of thermal and/or kinetic energy, two parametrizations of delayed cooling feedback and a 'mechanical' feedback scheme that injects the appropriate amount of momentum depending on the relevant scale of the SN remnant (SNR) resolved. All schemes make use of individually time-resolved SN events. Adopting isolated disk galaxy setups at different resolutions, with the highest resolution runs reasonably resolving the Sedov-Taylor phase of the SNR, I demonstrate that the mechanical scheme is the only physically well-posed method of those examined, is efficient at suppressing star formation, agrees well with the Kennicutt-Schmidt relation and leads to converged star formation rates and galaxy morphologies with increasing resolution without fine tuning any parameters. However, I find that it is difficult to produce outflows with high enough mass loading factors at all but the highest resolution. I discuss the various possible solutions to this effect, including improved modelling of star formation. Moving on to a more self-consistent setup, I carry out a suite of cosmological zoom-in simulations of low mass haloes at very high resolution, performed to z = 4, to investigate the ability of SN feedback models to produce realistic galaxies. The haloes are selected in a variety of environments, ranging from voids to crowded locations. In the majority of cases, SN feedback alone has little impact at early times even in low mass haloes ($\sim10^{10}\,\mathrm{M_\odot}$ at z = 0). This appears to be due largely to the build up of very dense gas prior to SN events, suggesting that other mechanisms (such as other stellar feedback processes) are required to regulate ISM properties before SNe occur. The effectiveness of the feedback also appears to be strongly dependent on the merger history of the halo. Finally, I describe a new scheme to drive turbulence in isolated galaxy setups. The turbulent structure of the ISM very likely regulates star formation efficiencies on small scales, as well as affecting the clustering of SNe. The large range of potential drivers of ISM turbulence are not fully understood and are, in any case, unlikely to arise ab initio in a whole galaxy simulation. I therefore neglect these details and adopt a highly idealised approach, artificially driving turbulence to produce an ISM structure of my choice. This enables me to study the effects of a given level of ISM turbulence on global galaxy properties, such as the fragmentation scale of the disk and the impact on SN feedback efficiencies. I demonstrate this technique in the context of simulations of isolated dwarfs, finding that moderate levels of turbulent driving in combination with SN feedback can produce a steady-state of star formation rates and global galaxy properties, rather than the extremely violent SN feedback that is produced by a rapidly fragmenting disk.