  Controlling physical systems and their dynamics on the level of individual
quanta propels both fundamental science and quantum technologies. Trapped
atomic and molecular systems, neutral and charged, are at the forefront of
quantum science. Their extraordinary level of control is evidenced by numerous
applications in quantum information processing and quantum metrology. Studying
the long-range interactions between these systems when combined in a hybrid
atom-ion trap has lead to landmark results. Reaching the ultracold regime,
however, where quantum mechanics dominates the interaction, e.g., giving access
to controllable scattering resonances, has been elusive so far. Here we
demonstrate Feshbach resonances between ions and atoms, using magnetically
tunable interactions between $^{138}$Ba$^{+}$ ions and $^{6}$Li atoms. We tune
the experimental parameters to probe different interaction processes - first,
enhancing three-body reactions and the related losses to identify the
resonances, then making two-body interactions dominant to investigate the ion's
sympathetic cooling in the ultracold atomic bath. Our results provide deeper
insights into atom-ion interactions, giving access to complex many-body systems
and applications in experimental quantum simulation.
