England defender Lucy Bronze took the women's award, seeing off the challenge of Wendie Renard and Pernille Harder.
Bronze, 29, who succeeds last year's winner Megan Rapinoe, won the Champions League with Lyon in August before moving back to England to play for Manchester City.
She was a suprise winner ahead of her former Lyon teammate Renard, and striker Harder, who moved from Wolfsburg to Chelsea in early September.
"What a surprise, to even be nominated alongside the two other players, they are both incredible players and outstanding human beings well," said Bronze during the ceremony.
"I don't think I have the words right now to explain how I am feeling. If there is anything 2020 has taught us it is to appreciate every moment you have, never look too far ahead and live in the here and now.
"I will appreciate winning this award now more than ever and I will remember this moment for the rest of my life."
France and Lyon's Sarah Bouhaddi capped her fine career with The Best FIFA women's goalkeeper award. Neuer won the men's award.
Liverpool manager Jurgen Klopp was named men's coach of the year while current Dutch women's national team boss Sarina Wiegman took the women's award for the second time in her career following her 2017 success.
The Puskas Award for best goal went to Tottenham's South Korean striker Son Heung-min for a mazy, individual goal he scored against Burnley in the Premier League in December 2019.
The Best awards began in 2017 after a brief merger with the Ballon D'Or, which was cancelled this year because of coronavirus, and the FIFA Player of the Year award, which was created in 1991.
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