Trump is accused of abuse of power and obstruction of Congress arising from his dealings with Ukraine.
U.S. Chief Justice John Roberts took an oath to preside over the trial and then swore in the assembled senators who will serve as jurors, asking if they would “do impartial justice”.
The 99 senators present signed their assent one by one.
Opening statements are scheduled to start on Tuesday.
Lev Parnas, Rudy Giuliani's associate, provided text messages and hand-written notes that add more details about the push for an investigation into Bidens.
According to the note, Parnas's chief objective in Ukraine was for its president to announce a criminal probe into Democratic frontrunner Joe Biden and his son.
Related: Trump demands immediate Senate trial