text: Djokovic, 29, swept past seventh seed Tomas Berdych 6-3 7-5 6-3 as he bids to win his first title at Roland Garros and complete the career Grand Slam.
  Williams, 34, was given a severe test by Kazakhstan's Yulia Putintseva, coming through 5-7 6-4 6-1.
  The American is trying to win her 22nd Grand Slam title and equal Steffi Graf.
  She will play Kiki Bertens in the last four on Friday, after the unseeded Dutch player saw off Swiss eighth seed Timea Bacsinszky 7-5 6-2.
  Djokovic will face Austrian 13th seed Dominic Thiem, who beat Belgian 12th seed David Goffin 4-6 7-6 (9-7) 6-4 6-1 to reach his first Grand Slam semi-final.
  Britain's Andy Murray will take on Stan Wawrinka of Switzerland in the other men's semi-final at 14:00 BST.
  Williams looked out of sorts for much of her quarter-final on a gloomy afternoon, with a sparse crowd on Court Philippe Chatrier.
  The champion was in deep trouble after an error-strewn set and a half, with the 5ft 4in Putintseva more than holding her own from the baseline and moving much the better in heavy conditions.
  Williams made 24 unforced errors in the first set to just two from her opponent, who had never been past the third round of a Grand Slam before this week.
  Putintseva then came back from 4-1 down in the second set to earn a break point at 4-4 that could have seen her serve for victory, but Williams held on and finally began to exert some control.
  She worked her way to a 5-0 lead in the decider, only for Putintseva to show one more flash of defiance by saving three match points before finally succumbing.
  "She's unbelievable and I didn't think I was going to win that but somehow I did," said Williams.
  Djokovic had won 23 of his previous 25 matches against Berdych of the Czech Republic, and was never seriously threatened after beginning their 26th meeting in commanding fashion.
  Berdych appealed for officials to take action when the rain, which has dogged this tournament, returned - but the closest Djokovic came to a quarter-final exit was a potential disqualification in the third set.
  He let his racquet slip out of his hand while angrily swiping at the ground and was relieved to receive just a code violation as it narrowly missed a line judge.
  "I am aware that I have been lucky," he said. "It was just some unfortunate bounce, but fortunate ending of the scenario. That's all I can say."
  Djokovic was already two sets up after dominating for an hour and a half, and moments later he recovered Berdych's early break of serve in the third set.
  Berdych described the tournament director's hesitance to halt play when the rain became heavier as "an absolute circus", but the delay lasted a little over 10 minutes and Djokovic soon wrapped up victory on the resumption.
  BBC tennis correspondent Russell Fuller
  "Djokovic has coped very well with the hand he has been dealt this week. Playing for the third day in a row, and in the knowledge he will be back in action on Friday, he was very sharp throughout against Berdych. Djokovic was fortunate, too, that the line judge saw his racquet coming and stepped out of the way, as he could have been disqualified had the racquet made contact.
  "Williams was moving very sluggishly in her quarter-final. Many spectators were wrapped up in blankets to protect themselves from the chill, and the three other women left in the draw should take great heart from the way the sparky Putintseva put the world number one under so much pressure."
target: Top seeds Novak Djokovic and Serena Williams remain on course to make history as they both reached the semi-finals at the French Open.