Seven-time major champion Williams, already a finalist this year at the Australian Open and Wimbledon, beat Kvitova 6-3 3-6 7-6 (7-2), clinching victory in a superb final-set tiebreak.
Below is all the information you need to know to watch Tuesday's action as well as a full list of men's and women's singles matches.
"If you are so in awe of someone being No. 1 in the world, then I don't think you're going into a match all focused and all there, ready to go", Vandeweghe said.
Kvitova returned to the tour in May at the French Open, where she made the second round while she also reached the same stage at Wimbledon.
"Was a great fight", Kvitova, 27, said Sunday after she uprooted Garbine Muguruza, the Wimbledon champion who had gobbled up momentum here. The last of the two wins in NY coming in 2001.
Williams reached this year's Wimbledon final before falling to Muguruza.
"It's such a wonderful feeling", she said of the tight win. Now the record is held by Serena Williams who won a Major at the age of 35.
Talking to the media after the match, William said, "I'm so fortunate to have won that match".
Then there is 24-year-old Sloane Stephens, who is playing in just her third-ever major quarterfinal since turning pro in 2010.
The 37-year-old's odds have tumbled through each round she has progressed, and before her quarter final match she was still available at 14/1 with one bookmaker.
"Life is still a little bit odd, but it's getting better as well", she admitted this week.
"My head is a little clearer, if that makes any sense", Stephens said.
"Reflecting back on the week, I can be happy", Sharapova said.
That thrilling five-set triumph ended Federer's five-year reign as champion and his 40-match win streak. This may finally prove to be her year, beating two seeded players and dispatching Vinci and Barty in straight sets.
The 83rd-ranked American has won 13 of her past 15 matches, all on hard courts, reaching the semifinals at three consecutive tournaments for the only time in her career. Stephens was a 7-6 (5), 6-1 victor over Williams in their lone prior encounter at the 2015 Roland Garros and carries great respect for the American icon into their rematch. "Seeing all the American players in the draw and all of them advancing so deep and competing so well", Williams said. "I wasn't trusting in it because I was kind of afraid because I didn't want to fall".
"Everything she's gone through".
She had an operation in January, and while forced to stay away from her sport, she found a new appreciation for it.
"She has something special which the other players don't have. I wanted to be doing something about my future". "I think it's dramatic fashion and it's great to see both of us go through".
"When situations get hairy, it's insane, I try to tell myself enjoy the competition, enjoy the battle", Williams told fans afterward. I play tennis and I have fun every day. "I just knew it would be about a few points".
"I will try my best".
One last shining, at her home, at her turf.
Earlier today, Stephens battled back from 1-3 down in the decider slipping past Anastasija Sevastova, 6-3, 3-6, 7-6 (4). She is looking to win her 50th WTA singles championship.
- 'I'm not a robot.