The first extraterrestrial radio source, detected by Karl Jansky, was the synchrotron radiation of cosmic ray (CR) electrons, associated with star formation, in the Milky Way's magnetic field. Our observational capabilities for studying these CRs in our own and other galaxies have grown vastly since then, both in radio and now gamma-rays. I will present my work on the theory of these CRs and the radiation they emit. I will describe a "conspiracy" that may set the radio luminosity of starbursts and its implications for the empirical "far-infrared--radio correlation" of star-forming galaxies. I will also discuss the implications of the recent gamma-ray detections of starburst galaxies on how CR protons are transported in them, how CR electrons cool in starbursts, and the magnetic fields in these galaxies.