"For every single day that I'm here I'm trying to do my best. This is only my fourth tournament back", said Williams, who has won all three of her previous meetings with Giorgi. "I'm just here just to be here and to prove that I'm back. I still have a long way to go to be where I was".
Ominiously for the other seven women left in the draw, the 23-time Grand Slam champion, who missed Wimbledon past year but won the title on her previous two visits, insists there is still plenty of room for improvement.
That is going to change now. After losing the first set, Williams was able to turn her game around and defeat Giorgi.
The key game was when Williams powered to a 2-1 lead in the decisive set, against the serve, without conceding a point.
It may be that today, for the first time, Serena Williams does not manage to intimidate her opponent before a ball is struck.More news: Jamie Vardy sends Gareth Southgate update ahead of World Cup semi-final
She probably should have done her homework on Williams, who crushed the upset-minded Italian after dropping the first set.
They clapped and yelled when Federer's drop volley capped a 16-minute tour de force of a first set in which he won 21 of the first 23 points.
It's okay, Serena, just keep your eyes on the prize at Wimbledon!
Kerber took advantage of Kasatkina's erratic serving and errors to return to the last four.More news: George Clooney recovering after scooter accident in Sardinia
Williams looked to be in a spot of bother when she surprisingly lost the opening set 6-3 and struggled with consistency from the back of the court. The reprieve was only momentary for Hsieh, as Cibulkova regained her composure to break and take the first set 6-4. But despite how close she is to another title and the record books, Williams is insistent that she doesn't need to win here in order to feel like her comeback has gotten off to a successful start. Nadal reached his first Wimbledon quarterfinal since 2011, winning in straight sets, and will have to beat Juan Martin del Potro or Gilles Simon to make the semifinals.
Kerber is a former No. 1 who owns two Grand Slam titles and was the runner-up to Williams at Wimbledon two years ago.
In a match that featured eight breaks of serve - four in the first five games - Ostapenko was able to elevate her game in the crucial moments.
Ostapenko played her usual aggressive style, compiling a 33-6 edge in winners. That top seeds from the women's side happened now shows that the field is quite even, maybe as well, regardless of who is 1, 2, 3, and 4.More news: Conte takes Chelsea training amid doubts over future