Caroline Wozniacki looked dominant in her first Grand Slam match since a stunning second-round exit at Wimbledon, defeating Samantha Stosur 6-3, 6-2 on Tuesday. With Simona Halep, the No. 1 player in the world, already eliminated, the women's draw at the US Open is a little more open for Wozniacki, who won her first career major at this year's Australian Open. She will try to keep that momentum going throughout the bottom half of the bracket, though Serena Williams looms large on the other side after a commanding first-round win.

No. 20-seeded Naomi Osaka is just 20 years old, but she's raring to take the next step in her career. Osaka made the fourth round in the Australian Open and the third rounds of both the French Open and Wimbledon, losing to Halep, Madison Keys and Angelique Kerber. The Japanese star won her opening match Tuesday, as she tries to continue her success.

No. 5 Petra Kvitova and No. 10 Jelena Ostapenko were also in action early, with Kvitova rolling and Ostapenko surviving a scare. 

Wimbledon winner Angelique Kerber also advanced with a sweep.

CoCo Vandeweghe, who made it to the 2017 US Open semifinals, was bounced in a performance that saw 12 double faults, eventually falling to Kirsten Flipkens.

On the men's side, high seeds Roger Federer, Alexander Zverev and Novak Djokovic all made relatively easy work of their first round opponents. The second-ranked Federer took down Japan's Yoshihito Nishioka in three straight sets. Federer improves to 18-0 in opening matches at the US Open.

The fourth-ranked Zverev rolled against Canada's Peter Polansky, taking him down in three dominant sets. The 21-year-old Zverev is the youngest player in the ATP Top 10.

Djokovic lost one set to his opponent, Marton Fucsovics, but finished strong to secure the victory and move on to the Round of 64. 

Here are some of the matches to watch out for on Tuesday as we update results throughout the day. 

Men's results

  • No. 2 Roger Federer defeats Yoshihito Nishioka 6-2, 6-2, 6-4
  • No. 4 Alexander Zverev defeats Peter Polansky 6-2, 6-1, 6-2
  • No. 6 Novak Djokovic defeats Marton Fucsovics 6-3, 3-6, 6-4, 6-0
  • No. 12 Pablo Carreno Busta defeats Malek Jaziri 7-5, 6-2, 6-2
  • No. 26 Richard Gasquet defeats Yuichi Sugita 6-3, 6-1, 6-3
  • No. 30 Nick Kyrgios defeats Radu Albot 7-5, 2-6, 6-4, 6-2
  • Joao Sousa defeats Marcel Granollers 6-2, 6-2, 6-3
  • Mikhail Kukushkin defeats Noah Rubin 6-3, 6-1, 4-6, 7-6
  • Jan-Lennard Struff defeats Tim Smyczek 7-6, 6-4, 6-3
  • Hubert Hurkacz defeats Stefano Travaglia 6-2, 2-6, 7-6, 3-0 (retired)
  • Tennys Sandgren defeats Viktor Troicki 6-3, 6-4, 6-2
  • Laslo Dere defeats Leonardo Mayer 6-4, 6-4, 4-6, 2-1 (retired)

Women's results

  • No. 2 Caroline Wozniacki defeats Samantha Stosur 6-3, 6-2
  • No. 4 Angelique Kerber defeats Margarita Gasparyan 7-6, 6-3
  • No. 5 Petra Kvitova defeats Yanina Wickmayer 6-1, 6-4
  • No. 10 Jelena Ostapenko defeats Andrea Petkovic 6-4, 4-6, 7-5
  • No. 13 Kiki Bertens defeats Kristyna Pliskova 6-0, 7-5
  • No. 14 Madison Keys defeats Pauline Parmentier 6-4, 6-4
  • No. 20 Naomi Osaka defeats Laura Siegemund 6-3, 6-2
  • No. 22 Maria Sharapova defeats Patty Schnyder 6-2, 7-6
  • Kirsten Flipkins defeats No. 24 CoCo Vandeweghe 6-3, 7-6
  • Lesia Tsurenko defeats Alison Van Uytvanck 6-3, 6-2
  • Aleksandra Krunic defeats Timea Bacsinszky 6-3, 3-6, 6-0
  • Julia Glushko defeats Monica Niculescu 3-6, 7-5, 6-4
  • Francesca Di Lorenzo defeats Christina McHale 6-1, 7-6
  • Wang Yafan defeats Anna Karolina Schmiedlova 6-1, 3-6, 6-4
  • Johanna Larsson defeats Alize Cornet 4-6, 6-3, 6-2
  • Taylor Townsend defeats Amanda Anisimova 3-6, 6-4, 6-3
  • Hsieh Su-wei defeats Ekaterina Alexandrova 6-3, 4-6, 6-3