We review recent advances in the numerical modeling of turbulent flows and star formation. An overview of the most widely used simulation codes and their core capabilities is provided. We then examine methods for achieving the highest-resolution magnetohydrodynamical turbulence simulations to date, highlighting challenges related to numerical viscosity and resistivity. State-of-the-art approaches to modeling gravity and star formation are discussed in detail, including implementations of star particles and feedback from jets, winds, heating, ionization, and supernovae. We review the latest techniques for radiation hydrodynamics, including ray tracing, Monte Carlo, and moment methods, with comparisons between the flux-limited diffusion, moment-1, and variable Eddington tensor methods. The final chapter summarizes advances in cosmic-ray transport schemes, emphasizing their growing importance for connecting small-scale star formation physics with galaxy-scale evolution.
C.F. acknowledges funding provided by the Australian Research Council (Discovery Projects DP230102280 and DP250101526), and the Australia-Germany Joint Research Cooperation Scheme (UA-DAAD). S.O. acknowledges support from NSF AST-2107340, NSF AST-2107942, NASA 80NSSC23K047, NSF AAG-2407522, a Peter O'Donnell Distinguished Researcher Fellowship, and a Donald Harrington Fellowship. We further acknowledge high-performance computing resources provided by the Leibniz Rechenzentrum and the Gauss Centre for Supercomputing (grants pr32lo, pr48pi and GCS Large-scale project 10391), the Australian National Computational Infrastructure (grant ek9) and the Pawsey Supercomputing Centre (project pawsey0810) in the framework of the National Computational Merit Allocation Scheme and the ANU Merit Allocation Scheme.