Context. Many galaxies contain magnetic fields supported by galactic dynamo action. The investigation of these magnetic fields can be helpful for understanding galactic evolution; however, nothing definitive is known about magnetic fields in ring galaxies. Aims. Here we investigate large-scale magnetic fields in a previously unexplored context, namely ring galaxies, and concentrate our efforts on the structures that appear most promising for galactic dynamo action, i.e. outer star-forming rings in visually unbarred galaxies. Methods. We use tested methods for modelling α-Ω galactic dynamos, taking into account the available observational information concerning ionized interstellar matter in ring galaxies. Results. Our main result is that dynamo drivers in ring galaxies are strong enough to excite large-scale magnetic fields in the ring galaxies studied. The variety of dynamo driven magnetic configurations in ring galaxies obtained in our modelling is much richer than that found in classical spiral galaxies. In particular, various long-lived transients are possible. An especially interesting case is that of NGC 4513, where the ring counter-rotates with respect to the disc. Strong shear in the region between the disc and the ring is associated with unusually strong dynamo drivers in such counter-rotators. The effect of the strong drivers is found to be unexpectedly moderate. With counter-rotation in the disc, a generic model shows that a steady mixed parity magnetic configuration that is unknown for classical spiral galaxies, may be excited, although we do not specifically model NGC 4513. Conclusions. We deduce that ring galaxies constitute a morphological class of galaxies in which identification of large-scale magnetic fields from observations of polarized radio emission, as well as dynamo modelling, may be possible. Such studies have the potential to throw additional light on the physical nature of rings, their lifetimes, and evolution.