Accusations that President Trump shaped American policy toward Ukraine for his own political gain exploded into a crisis for the White House in September after a whistle-blower complaint. Lawmakers have been examining how Mr. Trump pushed the Ukrainians to investigate the hacking of the Democratic National Committee in 2016 and former Vice President Joseph R. Biden Jr. and his son Hunter, a lawyer and former lobbyist. Those questions are now at the center of the House’s impeachment inquiry.

Once members of Congress turned their attention to Mr. Trump’s dealings with Ukraine, incorrect statements about the former vice president and his son started circulating again in far-right corners of the internet and on social media platforms like Facebook. Mainstream conservative media outlets like Fox News began running with some of the theories and Mr. Trump echoed them.

Here are four of the most prominent false or misleading rumors that have spread and an explanation of what really happened (to the extent that we know) .

Why was Ukraine’s top prosecutor fired?

The allegation at the heart of the controversy is that while Mr. Biden was vice president, he pushed to have Ukraine’s top prosecutor removed for investigating a company connected to Mr. Biden’s son Hunter, the Ukrainian natural gas firm Burisma. Videos pushing this theory began appearing on Twitter in late September and early October and have been viewed tens of millions of times. Mr. Trump’s campaign has also asserted the claim in ads on Facebook.