I recently moved into a new home and I am getting mold on my bathroom walls and ceiling. It appears that the last tenants have just painted over the mold, so it keeps coming back. Is there anything I can do to correct this?

Yes. Identify the source (for example a leak) of moisture/water and rectify that. Mold will only grow if there enough water to support growth. Moisture from cooking, dishwasher, shower and laundry is enough to support growth of some molds. Make use of fans in the kitchen and shower and if necessary use dehumidifiers.