Extragalactic surveys in the 2020s will reveal the full diversity of the galaxy assembly process: from environment-dependent evolution to the build-up of mass inside galaxies, and with a complete accounting of all relevant processes/constituents ensured by multi-wavelength coverage. Observations at radio wavelengths carry a unique potential in that they can probe star-formation activity and cold gas content, i.e. place constraints on both galaxy growth rates and the fuel for future galaxy growth. With the construction of the Square Kilometre Array (SKA), radio astronomy will enter a new era; high-sensitivity, high-resolution and dust-unbiased radio imaging will provide a confusion-free census of star formation and black hole activity, regardless of whether these occur in dust-obscured or optically thin regions. I will review the high-priority science cases developed by the SKA extragalactic continuum science working group and outline the challenges brought about by our still only partial understanding of some of the astrophysical mechanisms producing radio emission (e.g. the relation between star formation rate and radio continuum luminosity).
Thanks to an order of magnitude increase in survey speed compared to existing radio telescopes, SKA surveys will detect millions of galaxies. I will summarize how - in addition to advancing our understanding of the drivers of the cosmic star-formation history - commensal survey design will ensure that information on galaxy shapes, redshifts, spatial distribution and polarization can be used for cosmological studies and to trace the evolution of magnetic fields in galaxies and the cosmic web.