When the election returns came in Tuesday evening, the tallies for governor were razor-close, with the Democratic challenger, Attorney General Andy Beshear, ahead of the Republican incumbent, Gov. Matt Bevin, by 5,086 votes out of more than 1.4 million cast. Mr. Beshear claimed victory, but Mr. Bevin has refused to concede defeat.

Now what? How will Kentucky settle who will be governor for the next term?

The state’s Constitution and laws provide for several measures to decide a close or disputed election, and experts say that in the end, the voters may not have the final say. Here are the basics.