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The control of physical systems and their dynamics on the level of individual quanta underpins 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^,. Studies of the long-range interactions between these systems when combined in a hybrid atom–ion trap^, have led to landmark results^{, , , , , , , , , –}. However, reaching the ultracold regime—where quantum mechanics dominates the interaction, for example, giving access to controllable scattering resonances^,—has so far been elusive. Here we demonstrate Feshbach resonances between ions and atoms, using magnetically tunable interactions between ¹³⁸Ba⁺ ions and ⁶Li atoms. We tune the experimental parameters to probe …