In recent experiments with cylindrical and annular Hall thrusters, the enhanced cross-field electron transport induced by the rotating spoke was directly measured using a segmented anode. Approximately 50% of the total current was found to pass through the spoke. For the cylindrical Hall thrusters, the spoke oscillations were characterized using emissive and biased electrostatic probes and high speed imaging. The findings revealed a perturbed electric field which enhances electron transport across the field. Control and suppression of the spoke can be achieved by varying the effective boundaries of the thruster discharge from its cathode and anode sides. This includes recently demonstrated spoke suppression with a feedback control at the segmented anode. The magnetic field configuration of the thruster and the background gas pressure in the vacuum vessel can also affect the spoke oscillations.