Joe Biden's Senate seat is up for reelection this year. Now that he will be the Democratic vice-presidential nominee, does he need to give up his Senate seat? That is a matter of state law. According to various media reports today, Delaware law allows Biden to run for senator and vice president on the same day.
Lyndon Johnson faced the same scenario in 1960. Texas law was changed to allow him to run for both offices simultaneously. He won both elections, and subsequently resigned from the Senate.
That turned out to be a more important consideration for another Texan, Lloyd Bentsen, the unsuccessful Democratic vice-presidential nominee in 1988. "Lyndon's Law" was still in force in 1988. Bentsen was reelected to the Senate on the same day that his ticket headed by Michael Dukakis lost the presidential election. Bentsen continued to serve in the Senate until 1993, when President Clinton appointed him secretary of the treasury.
UPDATE: The Associated Press now reports that Biden will run for both senator and vice president.