Professorship without PhD in the United States
At most US institutions, you need the terminal degree in your field. For many fields this is a PhD, but in some it might be an EdD, a DMA, a DPH, a ThD, etc. However, all of these degrees are considered to one level or another to be research doctorates. Presumably faculty positions at a medical school require an MD or equivalent. Likewise for other professional schools. Positions in the visual arts, theater, dance, creative writing, cinematography, etc., may only require the MFA (Master of Fine Arts) degree, as it is considered to be the terminal degree in those fields. 

"Regulations" about the requirements to hold certain academic ranks and perform certain academic duties (like mentor graduate students) are made at the institutional level.