Worldbuilding: Putting the Civil in Civilization

The real test of a civilization is not just its laws, but how its laws are put into practice.  Criminal law is one aspect, of course: does the society care more about justice, or maintaining order?  Are trials designed to be fair, and truly find the guilty, or simply expedite a resolution.  Are the people protected from the state, or is the state protected from the people?

But civil law is another matter, and is often less examined in worldbuilding.  How do two citizens resolve a dispute or wrong in which, strictly speaking, no crime has occurred?  At least, in something resembling a culturally-accepted "official" way.  For some people, regardless of the level of civilization, the answer is, "Punch the other person in the face."  And at that point, quite possibly, a crime has occurred, or if nothing else, the situation will continue to escalate into more violence.

However, a society, in general, is going to want to avoid "punch person in the face" solutions in favor of non-violent, non-escalating one. 

Depending on how advanced the society is, this can take a lot of forms.  Sometimes it's just a ritual in which one party acknowledges they wronged the other, gives some form of recompense, and things return to normal.  However, the larger a society is, the more likely the two parties are complete strangers, and have little reason to trust each other to "do the right thing".  The right thing must be imposed, usually by contract or third-party arbitration.  Is this a formal or an informal process?  Is the third-party's authority universally recognized, or is simply agreed upon by the initial parties involved?

Now, in a lot of fantasy fiction, which often has its roots in an idealized version of medieval or Renaissance Europe, there is often the solution that is violent, but under strictly controlled rules and traditions.  I'm speaking, of course, of the duel.  But the duel does boil down to a formalized version of "punch the other person in the face" where it's designed to not escalate into further fighting, or even war. 

So, how does it work in your worlds, when no crime has been committed, but someone's honor must be satisfied, or a grievance must be addressed, or some other wrong righted?