How do they match up?published at 00:19 British Summer Time 6 September 2017
Williams v Kvitova
"One minute..." is the call from the umpire. It's almost time.
Let's see how these two Grand Slam winners' records shape up...
Williams beats Kvitova 6-3 3-6 7-6 (7-2) to reach semi-finals
Roof closed during second set on Arthur Ashe Stadium
Stephens beats Sevastova 6-3 3-6 7-6 (7-4) to make last four
Carreno Busta beats Schwartzman 6-4 6-4 6-2 to make men's semis
Alex Bysouth and Mike Henson
Williams v Kvitova
"One minute..." is the call from the umpire. It's almost time.
Let's see how these two Grand Slam winners' records shape up...
Williams v Kvitova
Gigi Salmon
BBC Radio 5 live sports extra commentator at Flushing Meadows
Venus Williams did not wave to the crowd as she came out, she has her game-face locked on. I have not seen her looking this psyched up for a while.
Williams v Kvitova
"Sometimes I just really feel that the touch is there - the strength, the aggressive kind of game plan," said Petra Kvitova, following her win over Wimbledon champion Garbine Muguruza in the last round.
Venus Williams, who saw off Carla Suarez Navarro, said: "I don't know how to explain my longevity. If I did I would bottle it and sell it. My sister and my family motivate me a lot.
"I am focused on myself and being as aggressive as possible. Nobody gives you a Slam, you got to take it and I am trying to take it."
Piers Newbery
BBC Sport at Flushing Meadows
Refuelling - or 'prefuelling', as I like to call it - is clearly vital before any elite sporting activity, none more so than a US Open nightshift. With that in mind, New Yorkers have sensibly been taking on board fuel for the battles to come, with Korean rice bowls edging out chicken burgers for the longest queue.
Perhaps of more significance to the actual tennis is the fact that it's now blowing a gale and the temperature is dropping as rapidly as the sun. Which makes sense. Since the roof was installed above the huge Ashe stadium it has been a bit more sheltered from the wind but it could still be a factor.
Williams v Kvitova
The players are out to warm up at the Arthur Ashe Stadium, how about a line from each?
Petra Kvitova: "It's what two players dream about, to stand under Ashe, and I'm looking forward to that."
Venus Williams: "It's so wonderful to see her here at her best. That's what everyone wants to see."
Williams v Kvitova
Gigi Salmon
BBC Radio 5 live sports extra commentator at Flushing Meadows
Petra Kvitova is quite a streaky player - but when it all comes together, as it did against Garbine Muguruza earlier this fortnight or against Eugenie Bouchard in the 2014 Wimbledon final, she is one of the best players in the world.
Very few can live with her.
Williams v Kvitova
This is Petra Kvitova's best performance at a Grand Slam since she reached the last eight in New York two years ago.
A lot has changed for the Czech since then.
The two-time Wimbledon champion severed tendons in all five fingers on her left hand during that knife attack nine months ago, and said last week it still did not feel completely back to normal.
The scars may still be visible, but through hard work and an awful lot of courage the 27-year-old has found her best form since returning to action in May.
And she's within three wins of a third major title.
Williams v Kvitova
Gigi Salmon
BBC Radio 5 live sports extra commentator at Flushing Meadows
It seems preposterous that Venus Williams could be still making the latter stages of Grand Slams at the grand old age of 37, especially considering she was diagnosed with Sjogren’s Syndrome back in 2011 - an auto-immune disorder that doesn't go away.
She must have so much passion for the sport.
The winner of this match will face Sloane Stephens in the semi-finals, after the American saw off Anastasija Sevastova 6-3 3-6 7-6 (7-4) in their last-eight encounter earlier.
"Making the semi-finals of my home Slam, it's indescribable," said the unseeded Stephens.
In the men's draw, Spain's Pablo Carreno Busta reached the US Open semi-finals for the first time with a straight-set win over Argentina's Diego Schwartzman.
The 12th seed took the match 6-4 6-4 6-2 in one hour and 58 minutes.
Meanwhile, Scotland's Jamie Murray and mixed doubles partner Martina Hingis are into the semi-finals at Flushing Meadows following a 6-3 2-6 10-8 victory over Abigail Spears and Juan Sebastian Cabal.
They will face Romanian Horia Tecau and women's singles quarter-finalist Coco Vandeweghe of the US in the last four.
Williams v Kvitova
Venus Williams has played more matches at Flushing Meadows than your average club player has hit winners at their local park court.
It's two decades since she first caught the attention at the US Open with a sensational run to the final, which eventually ended in defeat by Martina Hingis.
Eighteen visits to this corner of New York later and Williams is in the quarter-finals for a 12th time.
The seven-time major champion has overcome injuries and illness in recent years but, as one of only two Grand Slam champions left in the competition, could 2017 be her last shot at an eighth title?
It's 16 years since Venus Williams got her hands on the US Open trophy, but the American is the only player left in the women's draw to have the experience of doing so.
The 37-year-old, who was champion in 2000 and 2001, is looking for her first major victory in nine years.
First she'll have to successfully navigate the only other Grand Slam winner to have reached the quarter-finals, Petra Kvitova.
The Czech has made a remarkable return to the sport following surgery on her left hand after a knife attack last December.
Good luck choosing who to root for in this one.
Stephens 6-3 3-6 7-6 (7-4) Sevastova
Sloane Stephens takes her ball-blasting duties seriously, taking a long look up at the cheap seats to work out who deserves the celebratory fuzz most.
She will play the winner of Petra Kvitova and Venus Williams' match in the last four.
That Kvitova - Williams match is the next up on the main stage. We are expecting them under the lights at midnight BST.
And we will be here with live text and audio commentary.
Join us.
Murray/Hingis 6-3 0-3 Spears/Cabal
Jamie Murray and Martina Hingis won the first set of their mixed-doubles quarter-final against Abigail Spears and Juan Sebastian Cabal 6-3.
But they are 3-0 down in the second.
Murray and partner Bruno Soares earlier were knocked out of the men's doubles in a 6-1 6-2 defeat by Jean-Julien Rojer and Horia Tecau.
Stephens 6-3 3-6 7-6 (7-4) Sevastova
More from Sloane Stephens, who is the first of potentially four Americans into the women's semi-finals: "A month ago I was really worried about my protected ranking, not having enough tournaments and not being able to play.
"Once I realised my life is good, I play tennis every day, that relieved a lot of stress... and then bam! Here we are, semi-finals."
Stephens 6-3 3-6 7-6 (7-4) Sevastova
Here's Sloane Stephens: "Oh man, I'm getting teary-eyed. I think my coach is probably crying. This is just incredible, when I started my comeback at Wimbledon I could never have dreamed of this happening. Making the semi-final of my home Slam, it's indescribable.
"I just told myself to keep fighting. I'd been playing so well and so consistent, I knew if I stuck with it I'd have a chance, and that's what I did."
Stephens 6-3 3-6 7-6 (7-4) Sevastova
And what a way to do it!
Sloane Stephens grips and rips her backhand down the line, leaving Anastasija Sevastova a spectator.
Sloane Stephens breaks out into wreaths of smiles.
This is her second Grand Slam semi-final, but you have to go back to the 2013 Australian Open to find her first...
Stephens 6-3 3-6 6-6 (6-4) *Sevastova
Another bullet of a serve from Sevastova has Stephens dashing to her left, but the American finds a return.
It sits up for the Latvian, but no - she's found the middle of the net.
Two match points.
Stephens 6-3 3-6 6-6 (5-4) *Sevastova
Has that ball sneaked over the wire from Sloane Stephens' racquet?
Nope.
One mini-break back to Sevastova...
*Stephens 6-3 3-6 6-6 (5-3) Sevastova
Sloane Stephens takes a deep breath before her serve. The first catches the net. Let. But Anastasija Sevastova will challenge.
No such luck for the Latvian, much to the approval of this crowd.
It goes to Stephens's second serve though, and then she'll have to rally it out with Sevastova. A deep, looping backhand from the American sends Sevastova scrambling and she can only find the net with her return.
*Stephens 6-3 3-6 6-6 (4-3) Sevastova
Mini-break back to Sloane Stephens as Anastasija Sevastova goes for broke and comes up penniless, pushing her forehand wide.