Recent animal studies have provided new evidence that the cerebellum may have a stronger link to the reward system of the brain than was previously recognized. Direct projections from cerebellar deep nuclei (DN) to the ventral tegmental area (VTA) have been identified, and stimulation of these cerebellar afferents to the VTA was found to be rewarding. Such findings raise the possibility that cerebellar dysfunction could contribute substantially to addiction via a cerebellar influence over VTA. Consistent with animal findings, the investigators have found in human functional MRI (fMRI) preliminary data strong cerebellar and VTA activation in response to alcohol cues relative to non-alcohol stimuli in patients with alcohol use disorder (AUD) compared to controls, and close coupling observed between DN and VTA activation. Studying AUD and control participants, this project will address three important questions. The first is: What is the nature of cerebellar input to the VTA, and how is it perturbed in AUD? A number of investigations have suggested that when a stimulus is presented, the cerebellum generates a prediction of events that will follow based on prior associative learning, and then compares predicted and actual outcomes to generate a prediction error. The investigators hypothesize that these functions are disrupted in AUD. The investigators' preliminary data show that when an expected stimulus does not occur, a strong prediction error signal in the form of increased functional connectivity (FC) between cerebellum and its projection target is observed, and the investigators found an analogous increase in DN-VTA FC, that was abnormal in AUD patients, when alcohol pictures were presented. In Aim 1, using fMRI and a monetary incentive task, the investigators will investigate if DN-VTA FC reflects reward prediction and/or positive or negative reward prediction error. The second question is: Is the amount of activation in brain reward centers that is elicited by alcohol stimuli related to the amount of dysfunction in the cerebellum? In Aim 2 the investigators will investigate 2 measures of cerebellar integrity to determine the relationship with the magnitude of alcohol cue related activation in cerebellar, VTA, and other reward structures, and with DN-VTA FC: (1) The timing of the undershoot of the cerebellar hemodynamic response function (HRF), which has been found to be correlated with number of lifetime drinks; and (2) classical eyeblink conditioning, for which the cerebellum is necessary. The third question is: Can abnormal cerebellar activation and FC, as well as alcohol craving, be reduced by non-invasive cerebellar stimulation? In Aim 3, Using fMRI combined with cerebellar transcranial direct current stimulation (tDCS) during a cue reactivity task, the investigators hypothesize that in AUD participants cerebellar and VTA activation will be reduced, DN-VTA FC will be normalized, and alcohol craving will be reduced. The investigators will examine, using both resting state fMRI and psychophysiological interaction analysis, the effects of tDCS on FC among important structures of the reward system as well as on DN-VTA FC. These investigations will lead to a better understanding of the involvement of the cerebellum in AUD, as well as the therapeutic potential of cerebellar modulation.