Novak Djokovic, birthday boy Rafael Nadal and Roger Federer all reached the French Open third round on Thursday as the race for the women's title was thrown wide open after world number one Ashleigh Barty limped out.

Top seed Djokovic continued his bid for a second Roland Garros and 19th Grand Slam title with a straight-sets win over Uruguay's Pablo Cuevas.

The 34-year-old, who is looking to become the first man in more than 50 years to claim all four Slams multiple times, won 6-3, 6-2, 6-4.

"I stayed concentrated. I thought the third set was very difficult for me because he lifted the level of his game," said the world number one after his 350th Grand Slam match in which he fired 32 winners.

Djokovic will next face Lithuanian Ricardas Berankis.

World number three Rafael Nadal, bidding for a 14th French Open and record-setting 21st Grand Slam title, celebrated his 35th birthday by defeating Richard Gasquet, a player he first met when they were 12 years old.

Nadal stormed to a 6-0, 7-5, 6-2 win in a match played in an empty Court Philippe Chatrier as it clashed with a 9pm Covid-19 curfew.

Gasquet won just nine points in the first set. He managed a brief show of defiance from 2-5 to 5-5 in the second set but eventually fell to a 17th defeat in 17 matches with the Spaniard.

"The important thing is winning the matches. If it's in straight sets it's better," said Nadal. "I don't complain at all, the main thing is to feel myself playing well."

Next up for Nadal is Britain's Cameron Norrie who he has beaten twice this season, at the Australian Open and on clay in Barcelona.

Gasquet's loss means there are no Frenchmen or women in the third round for the first time.

Federer stayed on course for a quarterfinal showdown with Djokovic but only after becoming entangled in a rare running dispute with the chair umpire.

The 20-time major winner and 2009 champion in Paris saw off Marin Cilic for the 10th time in 11 meetings, winning 6-2, 2-6, 7-6 (7/4), 6-2.

Playing in only his second French Open since 2015, Federer goes on to face Germany's 59th-ranked Dominik Koepfer.

Women's top seed Barty, already suffering from a left hip injury, retired injured midway through the second set of her second-round match against Poland's 45th-ranked Magda Linette.

The Australian, who won the title on her last visit to Roland Garros in 2019, left the court for medical treatment after losing the first set 6-1.

She then called it quits at 2-2 in the second set on Court Philippe Chatrier. "It's heartbreaking," Barty said. "It's a miracle I got past the first round."

The 25-year-old's withdrawal leaves the tournament already without the world's top three ranked women's players.

World number two Naomi Osaka pulled out following a media boycott, saying she has suffered "bouts of depression" for three years.

Third-ranked Simona Halep, the 2018 French Open winner, withdrew before the event with injury.

Defending champion Iga Swiatek eased into the third round with a 6-1, 6-1 win over Sweden's Rebecca Peterson.

The eighth-seeded Pole goes on to face Estonian 30th seed Anett Kontaveit for a place in the last 16.

Swiatek hasn't dropped a set at Roland Garros in her last nine matches.

Spain's Carlos Alcaraz, 18, became the youngest player to make the last 32 since Andrei Medvedev in 1992 when he defeated 28th seed Nikoloz Basilashvili 6-4, 6-2, 6-4.

Two Italian teens made it through with 18th seeded Jannik Sinner seeing off compatriot Gianluca Mager while highly-rated Lorenzo Musetti put out Japan's Yoshihito Nishioka.

Elsewhere, Coco Gauff reached the third round for the first time as the 17-year-old American battled past China's Wang Qiang 6-3 7-6(1). 

Gauff eased through the opening set after jumping out to a 3-0 lead but did not have things her own way after that against the player she beat to win the Parma title last month.

Wang served for the second set at 5-3 but was broken and then had a set point on the Gauff serve after the American double-faulted.

But 24th-seeded Gauff showed great resilience to take the set into a tiebreak that she never looked like losing. After two straight-sets wins she will now face either Fiona Ferro or fellow American Jennifer Brady in the last 32.

Fourth seed Sofia Kenin, the 2020 runner-up, saw off American compatriot Hailey Baptiste 7-5, 6-3. Three-time quarterfinalist and fifth seed Elina Svitolina was similarly untroubled against Ann Li of the United States, winning 6-0, 6-4.

However, ninth seed Karolina Pliskova exited at the hands of 2018 runner-up Sloane Stephens as the 59th-ranked American won 7-5, 6-1.