Animals use feedback to rapidly correct ongoing movements in the presence of a perturbation. Repeated exposure to a predictable perturbation leads to behavioural adaptation that compensates for its effects. Here, we tested the hypothesis that all the processes necessary for motor adaptation may emerge as properties of a controller that adaptively updates its policy. We trained a recurrent neural network to control its own output through an error-based feedback signal, which allowed it to rapidly counteract external perturbations. Implementing a biologically plausible plasticity rule based on this same feedback signal enabled the network to learn to compensate for persistent perturbations through a trial-by-trial process. The network activity changes during learning matched those from populations of neurons from monkey primary motor cortex - known to mediate both movement correction and motor adaptation - during the same task. Furthermore, our model natively reproduced several key aspects of behavioural studies in humans and monkeys. Thus, key features of trial-by-trial motor adaptation can arise from the internal properties of a recurrent neural circuit that adaptively controls its output based on ongoing feedback.