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McDormand's win in Los Angeles on Sunday (local time) makes her a member of an elite club that includes Meryl Streep, Daniel Day Lewis and Jack Nicholson with three acting Oscars. The late Katharine Hepburn won a record four.
"My voice is in my sword. We know the sword is our work. And I like work," McDormand said in accepting the award, echoing a line from the film.
In Nomadland, the 63-year-old McDormand plays a widow who, in the wake of the US economic recession, turns her van into a mobile home and sets out on the road, taking seasonal jobs along the way.
The publicity-shy actress was one of the favourites for the best actress prize, which she first won in 1997 for her portrayal of a pregnant police chief in crime drama Fargo.
She won again in 2018 for playing an angry mother seeking justice for her dead daughter in dark comedy Three Billboards Outside Ebbing, Missouri.
The Oscar had been widely expected to go to the late Chadwick Boseman for his final film, Ma Rainey's Black Bottom.
The 83-year-old Briton has a six-decade film, television and stage career, but is perhaps best known for playing the brilliant but twisted murderer Hannibal Lecter in the 1991 thriller The Silence of the Lambs, for which he won his first Oscar.
His best lead actor win on Sunday made him the oldest actor to get an Academy Award - an honour previously held by the late Christopher Plummer.
In The Father, Hopkins plays an ageing man who has refused any help from his family and who is beginning to doubt what is real and what is imagined. It is adapted from a 2012 stage play of the same name. Hopkins told Variety that playing the role "made me very aware now how precious life is."
Born in Wales, the soft-spoken Hopkins is the son of a baker whose career has seen him playing characters ranging from the late US President Richard Nixon to artist Pablo Picasso, Pope Benedict and director Alfred Hitchcock. He was made a knight by the Queen in 1993.
The 39-year-old cemented her reputation as one of the most acclaimed directors in film with Nomadland, the story of financially stretched van dwellers that also won best picture.
It has been 11 years since Kathryn Bigelow became the first woman to win the director Oscar, for Iraq war drama The Hurt Locker.
It was the first Oscar for Zhao, who cast real-life nomads alongside McDormand to show the lives of older Americans who travel from job to job trying to scrape together a living.
Zhao was also nominated for best film editing and best adapted screenplay but did not win.
"I am extremely lucky to be able to do what I love for a living, and if this means more people get to live their dreams, I'm extremely grateful," she said backstage afterwards.
Zhao was born in China and lived in Beijing until age 14, when she went to boarding school in London and later finished high school in Los Angeles.
Britain's Daniel Kaluuya was named best supporting actor for his role as 1960s Black Panther activist Fred Hampton in Judas and the Black Messiah.
The movie Soul, the first from Pixar to feature a black lead character, won best animated feature.
MORE ORATORY, LESS MUSIC AND COMEDY
Social distancing forced a rethink of the ceremony, moving it to Union Station in downtown Los Angeles. After strict Covid-19 testing and quarantine protocols, nominees and their guests sat, mostly maskless, in a cabaret-style set or chatted in an outdoor courtyard.
The winners were chosen in a secret ballot by the 9000 members of the Academy of Motion Picture Arts and Sciences.
The Oscars was stripped to its bare essentials by constraints of the Covid-19 pandemic and aired with a look and feel like no other - devoid of the usual comedy and musical performances but chock full of lengthy oratory from the winners.
As promised in advance by producers, the 93rd annual Academy Awards ceremony immediately stood out as a sharp departure from televised Oscar presentations of the past, with no opening monologue or any of the glitzy song-and-dance numbers that typically fill the show.
Performances of the five Oscar-nominated best original songs were relegated to pre-recorded presentations broadcast during a two-hour pre-show ahead of the main event.
Coronavirus-related travel restrictions and public health measures forced a complete overhaul of the show, limiting attendance to just a few hundred nominees and presenters, with some contenders joining the festivities by satellite from international locations.
One of the greatest changes from traditional Oscar presentations during Sunday's telecast was a marked expansion in the length of speeches given by the winners, who normally are "played" off by the orchestra if their thank-yous run past 45 seconds.
Filmmaker Steven Soderbergh, producing the Oscars show this year with Stacy Sher and Jesse Collins, went out of his way to encourage each of the recipients to "tell a story" and personalise their acceptance speeches.
• BEST PICTURE
• BEST ACTOR
Anthony Hopkins - The Father
• BEST ACTRESS
Frances McDormand - Nomadland
• BEST DIRECTOR
Chloe Zhao - Nomadland
• BEST SUPPORTING ACTOR
Daniel Kaluuya - Judas and the Black Messiah
• BEST SUPPORTING ACTRESS
Youn Yuh-jung - Minari
• BEST ORIGINAL SCREENPLAY
Promising Young Woman
• BEST ADAPTED SCREENPLAY
• BEST ANIMATED FEATURE FILM
• BEST DOCUMENTARY FILM
My Octopus Teacher
• BEST INTERNATIONAL FEATURE FILM
Another Round (Denmark)
• BEST ORIGINAL SONG
"Fight for You" - Judas and the Black Messiah