Heavy crude oils having high sulfur content and viscosities are upgraded by a hydrodesulfurization (HDS) process that includes microwave irradiation of a mixture of the sour heavy crude oil with at least one catalyst and optionally, one or more sensitizers, and irradiation in the presence of hydrogen. The process is also adapted to microwave treatment of hard to break emulsions, either above ground or below ground where water-in-oil emulsions are initially formed, followed by the catalytic hydrodesulfurization promoted by application of further microwave energy to the demulsified crude oil stream.