December 5 has been declared a national day of mourning – many government offices and US stock exchanges are closed.
Following the funeral service, George H.W. Bush’s flag-draped casket was carried from the cathedral by a military honor guard and was later flown to Texas, arriving in Houston at 23:00 GMT.
The casket will lie in public repose at St Martin’s Episcopal Church until December 6.
There will be a second service before the Bush 4141 – a train which was named in honor of the president – takes his body to its final resting place, next to his wife Barbara at the George HW Bush presidential library.
Jim McGrath, who was George H.W. Bush’s spokesman, said the president had once wondered if anyone would attend his funeral.
George H.W. Bush was a decorated fighter pilot in World War Two, a head of the CIA and vice-president to Ronald Reagan before being elected president in 1988.
His time in office was dominated by foreign policy – and he was praised for how he navigated the end of the Cold War, and handled the first Gulf War.
However, he was accused of neglecting domestic affairs and, after reneging on a campaign promise not to increase taxes, lost to Bill Clinton in the 1992 presidential election.
George H.W. Bush has been described as a bipartisan figure who tried to make conservatism “kinder and gentler”.
His coffin had earlier left Texas to a 21-gun salute and was carried to Washington on board Air Force One – temporarily renamed Special Air Mission 41 in homage to the late president.
George H.W. Bush, who served as the 41st president between 1989 and 1993, had been receiving treatment for a form of Parkinson’s disease and had been admitted to hospital with a blood infection in April.
The public will be able to pay respects at the rotunda until December 5, when an invitation-only funeral service will take place at the National Cathedral.
President Trump will attend with the first lady but will not deliver a speech.
According to Mark Updegrove’s book The Last Republicans, which came out in 2017, George H.W. Bush voted for Donald Trump’s opponent Hillary Clinton in the election and had called him a “blowhard”.
Former President George W. Bush, worried he would be the “last Republican president”, despite Donald Trump running on the party’s ticket, the book said.
A service was held in the rotunda after George H.W. Bush’s coffin arrived on December 4, with members of the military serving as pallbearers. George W. Bush was in attendance and was clearly moved by the events.
In a eulogy, House Speaker Paul Ryan said: “Here lies a great man”, adding that “no-one better harmonized the joy of life and the duty of life”.
Paul Ryan said George H.W. Bush was a “great leader and a good man, a gentle soul of firm resolve”.
Senate majority leader Mitch McConnell said: “Through the Cold War and the Soviet Union’s collapse, he kept us on course. When the rule of law needed defending in the Persian Gulf, he kept us on course.”
Vice-President Mike Pence said there was a “kindness about the man that was evident to everyone who met him”.
George H.W. Bush was a decorated fighter pilot in World War II, a head of the CIA and vice-president to Ronald Reagan before being elected president in 1988.
Also making the final journey with President George H.W. Bush was his loyal labrador service dog, Sully.