Fifteen years Williams' junior, 15th-seeded Madison Keys appears ready to navigate the U.S. Open draw after being reunited with her coach Lindsay Davenport and looking fully recovered from wrist surgery late last year.
But this season, at age 37, Williams has turned back the clock, reaching the finals at Wimbledon and the Australia Open.
"I just want to win, and if you get the win, you get the ranking", said Williams, who was previously at No.1 in 2002.
Since Williams became the first unseeded US Open women's finalist since 1958, she has won the 2000 and 2001 US Opens and five Wimbledon crowns, including her most recent Slam title in 2008.
Venus Williams has reached the third round for the seventh Grand Slam tournament in a row.
"I had no idea what she looked like, who she was, anything".
'But she played unbelievable.
"She played fantastic. She played well, served well, competed well".
"Honestly, if my kid can grow up exactly how I did, I couldn't be happier", she says. The match will begin at 7 p.m. ET.
'I definitely wasn't expecting or planning on dropping sets today.
She was up a set, plus a break in the second at 2-0, when she faltered.
Muguruza is arguably the hottest player on the women's tour and is considered one of the favourites, despite never having been past the second round at Flushing Meadows.
Jared Donaldson and Steve Johnson also won through but 13th seed Jack Sock was not so fortunate, going down 6-2 7-6 (12) 1-6 5-7 6-4 to Australian Jordan Thompson.
Other matches involving seeded players to finish quickly during Monday's opening session included 18th seed Caroline Garcia of France defeating Czech Tereza Martincova 6-0 6-1 and 31st seed Magdalena Rybarikova of Slovakia defeating Italian Camila Giorgi 6-3 6-4. On the women's side, there as No. 21 Ana Konjuh, No. 24 Kiki Bertens and No. 32 Lauren Davis.
Sock, a two-time medalist at the Olympic Games Rio 2016, will be looking to make a Grand Slam quarterfinal for the first time; he made it to the fourth round of the U.S. Open for the first time in his career a year ago.
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