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Abstract Galaxies are observed to host magnetic fields with a typical total strength of around 15 $$\upmu $$ $\mu$G. A coherent large-scale field constitutes up to a few microgauss of the total, while the rest is built from strong magnetic fluctuations over a wide range of spatial scales. This represents sufficient magnetic energy for it to be dynamically significant. Several questions immediately arise: What is the physical mechanism that gives rise to such magnetic fields? How do these magnetic fields affect the formation and evolution of galaxies? In which physical processes do magnetic fields play a role, and how can that role be characterized? Numerical modelling of magnetized flows in galaxies is playing an ever-increasing role in finding those answers. We review major techniques used for these models. Current results strongly support the conclusion that field growth occurs during the formation of the first galaxies on timescales shorter than their accretion timescales due to small-scale turbulent dynamos. The saturated small-scale dynamo maintains field strengths at only a few percent of equipartition with turbulence. This is in contradiction with the observed magnitude of turbulent fields, but may be reconciled by the further contribution to the turbulent field of the large-scale dynamo. The subsequent action of large-scale dynamos in differentially rotating discs produces field strengths observed in low redshift galaxies, where it reaches equipartition with the turbulence and has substantial power at large scales. The field structure resulting appears consistent with observations including Faraday rotation and polarisation from synchrotron and dust thermal emission. Major remaining challenges include scaling numerical models toward realistic scale separations and Prandtl and Reynolds numbers. 

Abstract Magnetic fields are now widely recognized as critical at many scales to galactic dynamics and structure, including multiphase pressure balance, dust processing, and star formation. Using imposed magnetic fields cannot reliably model the interstellar medium's (ISM) dynamical structure nor phase interactions. Dynamos must be modeled. ISM models exist of turbulent magnetic fields using small-scale dynamo (SSD). Others model the large-scale dynamo (LSD) organizing magnetic fields at the scale of the disk or spiral arms. Separately, neither can fully describe the galactic magnetic field dynamics nor topology. We model the LSD and SSD together at a sufficient resolution to use the low explicit Lagrangian resistivity required. The galactic SSD saturates within 20 Myr. We show that the SSD is quite insensitive to the presence of an LSD and is even stronger in the presence of a large-scale shear flow. The LSD grows more slowly in the presence of SSD, saturating after 5 Gyr versus 1–2 Gyr in studies where the SSD is weak or absent. The LSD primarily grows in warm gas in the galactic midplane. Saturation of the LSD occurs due toα-quenching near the midplane as the growing mean-field produces a magneticαthat opposes the kineticα. The magnetic energy in our models of the LSD shows a slightly sublinear response to increasing resolution, indicating that we are converging toward the physical solution at 1 pc resolution. Clustering supernovae in OB associations increases the growth rates for both the SSD and the LSD, compared to a horizontally uniform supernova distribution.