Devices and methods provide for ablation of cardiac tissue for treating cardiac arrhythmias such as atrial fibrillation. Although the devices and methods are often to be used to ablate epicardial tissue in the vicinity of at least one pulmonary vein, various embodiments may be used to ablate other cardiac tissues in other locations on a heart. Devices generally include at least one tissue contacting member for contacting epicardial tissue and securing the ablation device to the epicardial tissue, and at least one ablation member for ablating the tissue. Various embodiments include features, such as suction apertures, which enable the device to attach to the epicardial surface with sufficient strength to allow the tissue to be stabilized via the device. For example, some embodiments may be used to stabilize a beating heart to enable a beating heart ablation procedure.