We will do everything we can to ensure that you . . . have what you need, Biden continued.
Biden "looks forward to a smooth transition", and Trump said it will be as "smooth as it can be".
Politics is tough, and in many cases it's not a particularly nice world. Today it's a nice world, and I appreciate it very much, Trump said after the two shook hands.
The meeting between the two lasted for nearly two hours.
Biden emphasized the importance of continued support to Ukraine, and conveyed "his view that the US's continued support to Ukraine is in our national security interest", according to Jake Sullivan, the US national security adviser, during a press conference.
Earlier in the day, Trump hinted that he is open to serving a third term as president, which is not allowed under current legislation.
I don't think I'll run again, unless you say "he's good, we need to come up with something else", he said.
Trump arrived in Washington in the afternoon, Swedish time, ahead of the meeting with Biden.
