There have been some surprising upsets in early rounds with the most notable being defending champion Jelena Ostapenko  losing in the very first round (in straight sets 7-5 6-3!) on the very first day of the tournament (which I attended). On the same day (a few hours before) Venus Williams  also lost in the 1st round, also in straight sets, 6-4 7-5, Qiang Wang. Other "early" round losers were Petra Kvitova  (l. Anna Kontaveit) and Elina Svitolina  (l. Victoria Kuzmova).
Serena Williams is playing in her first Grand Slam tournament since having a child and getting married (in that order :)) and although she had looked scratchy in her earlier tournaments this year (in Indian Wells and Miami) in Paris she has surprisingly looked quite solid and yesterday defeated Julia Georges  to make it to a 4th round meeting with Maria Sharapova who also looked very sharp in taking out Karolina Pliskova . It will be there 22nd meeting and Serena has won the last 18 times the two have played since 2004.
Simona Halep  and Garbine Muguruza  have looked the sharpest so far in the tournament but Caro Wozniacki , Sloane Stephens  and Madison Keys  are all dangerous contenders still in the draw.
The biggest seed on the men's side to fall in the early days of the tournament was Grigor Dimitrov  who lost in straight sets to 34-year-old Fernando Verdasco in the 3rd round. The story on the men's side has been the travails of Sascha Zverev  to live up to his outstanding ATP tour play and make a move in the majors. After cruising in his 1st round match by losing 4 games against Richar Berankis, Zverev has played three consecutive 5-set matches, and saved a match point in the 3rd round. The German is now in his first major quarterfinal.
Rafael Nadal is going for his 11th French Open title and it doesn't look like anyone will be able to stop him from doing so this year. He was forced into a tiebreak in his very first match with Simone Bolleli but since then the King of Clay has not given up more than 3 games in a set.
The other contenders remaining in the tournament are Dominic Thiem , Marin Cilic , Juan Martin del Potro , David Goffin  and John Isner .