It is commonly understood that romantic relationships between faculty and students at the same university are frowned upon, or explicitly forbidden. However are romantic relationships between faculty and graduate students at geographically far away universities acceptable? Are they still acceptable if the faculty and graduate student are in the same field or neighboring field? Is a small age difference between faculty and graduate student ever relevant in such situations?
Short answer: generally yes.
It is commonly understood that consenting adults are in general free to enter voluntary romantic relationships with whomever they choose, except for people over whom they hold professional authority. Thus, a professor may not have a relationship with her or his own student. That’s a big no-no anywhere I’m familiar with.
As for graduate students in the same institution but in other departments, whether a professor is to be regarded as being in a position of authority over such a student is a matter of interpretation; opinions about this seem to vary considerably, and I think it would be a mistake to characterize any particular opinion as being “commonly understood”.
As for a professor having a relationship with someone who is a student in another university, there may be specific circumstances where that would be problematic, such as if the student is applying for a postdoc or other position in the professor’s department. Even this doesn’t make the relationship taboo, it only means appropriate measures must be taken to ensure the professor isn’t exercising any decision-making authority over their significant other’s application. But generally speaking, other than those specific exceptional circumstances, this doesn’t violate any rules, and isn’t frowned upon by reasonable people.
Look for guidance to the reasons behind such rules and norms. They serve, formally and/or informally to prevent situations in which one party can blackmail or extort the other into improper acts that affect the institution. The same sorts of rules occur in many large organizations for the same reasons.
Can a History professor adversely affect the career of a philosophy graduate student? Perhaps they can, though it would be less likely in a very large university. But each case would need to be judged individually. They could also improperly work to advance the career of the other person in some situations.
However, most formal rules also have the purpose of trying to avoid the appearance of impropriety, not just the fact of it. Therefore they are normally written (when written) to be a bit more conservative than some would consider necessary. Scandal reflects badly on the institution, of course.
But a relationship between someone at one university and someone else at a different university is unlikely to have the power imbalance characteristics that would raise alarms if their professional association were closer.
So the answer to the question "Is it alright to ..." is normally "It depends...". But some cases are much clearer than others. And predators do exist.
I'm not as sure as some of the other answers here, although agree with @Buffy that "it depends". Many of the answers here are posted under the assumption that what is wrong with faculty-student relationships is that the faculty member has the ability unfairly advantage the student, but this isn't the only reason.
Part of the worry with staff student relationships is that the relationship is not entered into entirely voluntarily, even if the staff member believes it is (actaully, even if the student believes it is). This is because the staff member has power over the student - but not just concrete power, like grade etc, but also social and cultural power. This differential can also apply outside your own university - students can have a hard time dealing with attention from staff at conferences because the staff member has social capital in the field that the student does not have, thus there might be an element of coercion involved, even if neither party (but especially the staff member) is aware of it.
Thus, to a certain extent, common sense has to be used - relationships that pre-exist the current staff/student status, or two people who meet at say, a Star Trek conversion, without being award of the connection, are clearly fine. Relationships that start at conferences are much more dodgy. As @Buffy said, look at the reasons behind the rules, and ask your self how this applies to your situation.
I would carefully check with each school's ethic guidelines, which should be written somewhere. That said, in general, I'd be surprised if there were any issue. There might be some possible complicating factors; for example, some schools will occasionally have faculty from outside sit on a PhD committee, and obviously it would be inappropriate for that faculty member to sit on that grad student's committee. But barring weird situations where they end up having some conflict of interest or authority over the grad student, there really shouldn't be an issue.
A lot of these answers focus on the direct interpersonal aspects of the situation (eg, can two people date).
Here's a direct quote from my university's handbook about another aspect worth considering:
A potential conflict of interest exists whenever personal, professional, commercial, or financial interests or activities outside of the University have the possibility (either in actuality or in appearance) of (1) compromising a faculty or staff member’s judgment; (2) biasing the nature or direction of scholarly research; (3) influencing a faculty or staff member’s decision or behavior with respect to teaching and student affairs, appointments and promotions, uses of University resources, interactions with human subjects, or other matters of interest to the University; or (4) resulting in a personal or family member’s gain or advancement at the expense of the University
For example, this might cover:
Faculty member reviews manuscripts, grants, etc from the lab of the student
Faculty member uses official resources to travel to student's city for borderline reasons, or requests reimbursement for non-work-related costs during the trip (if you want to upgrade to the honeymoon suite, don't pay for it from a grant)
Unusual or preferential access to lab resources while student is visiting ("here's our visiting scholar. Give them whatever they need. Really.")
Sharing sensitive or privileged information (eg third-party manuscripts under review that might affect student's chances of publication).
None of these scenarios is specific to a relationship, but some of these situations might be more likely to occur when a romantic relationship is involved. All the standard rules of good judgment apply.