Alexander Zverev (Ger) bt Marin Cilic (Cro) 6-4 3-6 6-4
Roger Federer (Swi) bt Jack Sock (US) 6-4 7-6 (7-4)
You can listen to BBC Radio 5 live sports extra coverage of Alexander Zverev v Marin Cilic at the top of this page from 20:00 GMT.
Earlier, Roger Federer beat Jack Sock 6-4 7-6 (7-4) in the opening match of the ATP Tour finals and you watch the five best shots from that encounter below.
ATP Finals: Five best shots as Federer beats Sock in opener
Image source, Getty ImagesAnd that concludes our coverage on these pages for the day following Roger Federer's perfect start to his bid for a seventh ATP World Tour Finals crown.
Alexander Zverev takes on Marin Cilic later in the evening session in the opening match of the Boris Becker group. You can listen to that one on BBC Radio 5 live sports extra or at the top of this page.
We'll be back tomorrow evening for live-text commentary on Rafael Nadal v David Goffin while there will be BBC TV coverage of Dominic Thiem v Grigor Dimitrov tomorrow afternoon.
See ya!
More from Rafael Nadal: "I want to say thanks all of the tournaments over the year and all of the fans.
"You have been a huge support during the whole year in every event. Without all of you nothing would be the same - see you tomorrow on court."
Rafael Nadal speaking on court after being presented the year-end world number one trophy: "It has been a fantastic season, a very emotional one after all of the things I have been going through in the last couple of years with injury. To have this trophy again is something I never thought was possible.
"None of this would have been possible without my team and family. Many thanks for the support. There have been tough moments but we kept believing."
There have been some doubts over Rafael Nadal's fitness this week but he ends his on court interview by saying "see you on court tomorrow".
Good signs for this tournament.
Full quotes coming up...
Image source, ReutersSuited and booted, a very dapper Rafael Nadal strides onto the court.
He gets a huge reception from the crowd.
They've seen two of the greatest in the flesh today.
And now a bonus for this O2 Arena crowd.
Rafael Nadal is going to be awarded his prize on court for finishing as year-end world number one for the fourth time.
Image source, Getty ImagesRoger Federer: "It’s wonderful to be back (in the ATP Finals), especially after missing last year with injury. It was tough not to be here but at the same time I really enjoyed the battle for number one between Andy (Murray) and Novak (Djokovic).
"But it’s great to be back here, I love playing at the O2."
Roger Federer: "It was really just a matter of trying to keep the ball in play because it’s like the first round of any other tournament, where there’s always a struggle because you’re not quite used to the conditions, how the ball is flying.
"For a while you’re just happy to hit some good shots, focus on your own serve and have good intensity.
"I got off to a great start, my big hope was I was going to be able to play a bit more freely after that. The second set was tight, I missed some opportunities, the breaker could have gone either way and in the end he helped me with some double faults and some mistakes.
"I’m really happy that I got through somehow."
Federer 6-4 7-6 (7-4) Sock
Tim Henman
Former British number one on BBC TV
It was good performance [from Federer] and good performance from Jack Sock.
He didn't get off to the best start but that was the only break of the match.
In the second set Federer was creating the opportunities. He had five break points but Sock was able to hag in there.
He was dominant on serve and that proved to be decisive factor.
Federer 6-4 7-6 (7-4) Sock
A scribble of his name on the camera lens and a little wave.
Ever the charmer.
After missing this tournament last year, Roger Federer is back at the O2 with a win.
Image source, Reuters
Federer 6-4 7-6 (7-4) Sock
All over!
Jack Sock goes long and, despite a challenge, the result is confirmed.
A good effort from Jack Sock but Roger Federer was just serene on serve.
Image source, Getty Images*Federer 6-4 6-6 (5-4) Sock
No. no. no.
A double fault at the worst possible time from Jack Sock and he knows it.
A shake of the head. Federer has two serves now. Win them both and the match is his.
Image source, ReutersFederer 6-4 6-6 (4-3) Sock*
Federer clearly wants this over and done with but Jack Sock gets the break back after the change of ends!
Federer challenges the ball from Sock but the American's backhand found the baseline.
Back on serve.
*Federer 6-4 6-6 (3-2) Sock
A little grumble from Federer as he makes an error. He just can't quite wrap this up.
Angry Fed = brilliant Fed. A crunching backhand and now he does have the mini-break. That'll cheer him up.
Image source, EPAFederer 6-4 6-6 (2-1) Sock*
No he can't. Two mistakes from Jack Sock on the first Federer service points.
Fed back ahead.
*Federer 6-4 6-6 (0-1) Sock
Big forehand from Sock. He wins the first point of the breaker.
Now can he win a point on the Federer serve for only the third time in this set?
Tim Henman
Former British number one on BBC TV
Federer’s motivation is his love of tennis. It’s about the future, improving his game and enjoying the process. He enjoys all the different aspects of tennis, to travel as much as he does and compete.
Federer 6-4 6-6 Sock*
I said in that last game Jack Sock has been impressive at times in this set but just a couple of times on big points he's been loose.
He makes errors on the first two points of this Federer service game and the world number two flies through another game in little over a minute.
To a tie-break we go.
Image source, AFP*Federer 6-4 5-6 Sock
Fist pump from Jack Sock. The break point is saved with a couple of pounding forehands to the Federer backhand. When he's been good Jack Sock has been really good in this set.
Now a good use of hawkeye too. A serve is called out but Sock challenges and it's in! An ace!
The crowd cheer. The American looks to his box. This one isn't done yet.