Is there any specific rule that requires someone supervising and awarding a Ph.D. to have a Ph.D. themselves? I learned recently in some parts of Asia it is possible to have a Masters degree and award a Ph.D. One point made was, in the example of Korea, there was no education system and as such it was required for this to be the case at some point. This seems to have continued. Does this exist in United States or Europe as well?
No, there isn't any general rule.
First, a PhD degree is awarded by a university, not by an individual.
The education system of a country or the university rules usually define which positions can officially act as supervisors. The same education system or the university rules establish which titles should have a person to be eligible for those positions: if the PhD is not required, which is not uncommon, there can certainly be PhD supervisors without a PhD.
Another case is the following: in Italy the PhD degree has been established some thirty-odd years ago. This means that most of those who became professors in Italy before that time don't have a PhD (as Federico rightly observes, some might have taken it abroad), but they surely can supervise students officially.
Different systems are in place in different universities and countries to govern who can formally supervise PhD dissertations. This is sometimes called promotion rights. There are, as far as I know, two ways how one can conventionally acquire promotion rights:
- Due to holding a position that grants promotion rights, such as being a Full Professor in Germany, being an Assistant, Associate, or Full Professor in Switzerland or the United States, or being a faculty member on any level (I think?) in the UK. Note that, in some cases, promotion rights can also be negotiated even if the position per se does not grant them. I know a case in Austria of an Assistant Professor who negotiated promotion rights as part of his job package when accepting his post, even though Assistant Profs. typically do not have promotion rights at this university.
- Due to having habilitation or whatever the national equivalent is. This seems to be a bit of a historical residue primarily in central and northern Europe (e.g., Austria, Germany, Sweden, ...). Habilitation can be earned not unlike getting a PhD degree (with a thesis and a public defense), and pretty much by definition grants the holder not the job of a professor, but all the "rights" associated with being one (primarily advising graduate students and teaching courses and giving lectures). Historically, to become a professor in countries that had habilitation one was expected to get habilitation first. This is not necessarily the case anymore, but habilitation still exists. A person that has habilitation but no professor position is, in German, called a "Privatdozent" (private teacher). Such a person is able to advise PhD students, but without close collaboration with somebody within the university, the formal proceedings are cumbersome for all involved.
Note that both versions not necessarily require a PhD (one could for instance be appointed Full Professor without a doctorate, and there are indeed disciplines and cases where this happens). However, in practice, you can assume that all typical and realistic career paths that lead to promotion rights include doing a PhD yourself first.
I'll speak from my experience in research in France and in Canada.
As pointed out in other answers, the supervisor do not award the diploma themselves, it is the university that does, and it only happens after a comittee made of different people decide that the thesis and defence are worth a PhD. So, there are two goups of people whose diploma we can consider here: the supervisor(s) and the comittee.
Supervisors: it is fairly common to have more than one spervisor, officially or not. The official supervisor has to comply with official rules: in France, they need a PhD + a "HDR" (habilitation à diriger des recherches = authorization to supervise research, which you obtain with yet another thesis); in Canada, I think you need to have a PhD + be a professor. On the other hand, the daily supervisor (the one that interacts the most with the student) may be this official supervisor, but they may be someone else, and in this case I don't think there is any official rule, as I've seen professors, postdocs, or people with Masters having this role.
Comittee: the composition of the comittee depends on the countries and on the universities, but I'm fairly certain you need PhDs for at least part of the comittee members. In France for example, you need to follow strict rules with a certain number of professors, a certain number of people from outside you university, etc.
Andrew Casson, a renowned geometric topologist, never received his PhD. And the institutions where he worked have (gladly, and proudly, I suspect) bestowed degrees on his PhD students.