In the year 2000, the state of Florida was in the process of various recounts when Bush v Gore put an end to the process on Dec 12, 2000, which allowed George W. Bush to become the 43th President after being awarded Florida's electoral votes.
While unlikely, there is a distinct possibility that there could be enough contested states still fighting it out like that in 2020 (due to various court-mandated rule changes and close counts with disputed absentee/main-in balloting) that come mid-December there's no winner in the Electoral College to vote on. It's even possible these drag on through the courts and come Jan 20, 2021, there's no winner to inaugurate. This answer notes
In a nightmare scenario where the popular vote cannot be certified or cannot be certified in time, the realistic option would be for the state legislature to appoint a slate of Electoral College delegates instead. The date the Electoral College meets (December 14 this year) is specified by Federal law so that cannot realistically be moved. Federal law sets a "safe harbor" date (December 8 this year) which is the deadline for states to choose their slate of electors and ensure that Congress accepts them. It is highly unlikely that you could have an election November 3, count the ballots, work through the litigation that would undoubtedly arise if there was large-scale fraud or large-scale disenfranchisement due to riots or COVID or some other reason, schedule and hold a new election, count those ballots, and have the resulting slate of electors appointed by December 8. And if the issues are not with fraud but with an inability to completely count or recount the votes quickly enough because of delays due to COVID, states rushing to adopt widespread mail-in voting, court challenges, etc., starting a new election would be counterproductive. There are reports that both sides have already started game planning scenarios in which friendly legislatures or governors in swing states are called on to do just this.
That's fine, but I could easily see Federal courts jumping in and enjoining states where the outcome of one or more court cases will determine which ballots can and cannot be counted. So let's say Trump ends election night up by 500 votes in Michigan (which he needs to win), but 3000 Biden votes are ruled invalid (didn't properly sign, returned too late, etc.). There would be a Federal lawsuit and that lawsuit would undoubtedly enjoin the state legislature from appointing electors (and possibly postponing the Electoral College itself). As there's only 79 days between Election Day and Jan 20, that doesn't leave much time to resolve the status of states. Neither side seems likely to concede.
If there's no clear winner by noon, Jan 20, what happens?