Serena aims to catch legends Martina Navratilova, Chris Evert
Some were surprised that it took Williams till now to return to the top of the rankings considering the excellence of her run in the second half of 2012. During this she won two Grand Slam titles, the Olympics, and the WTA Championships season-end title, and captured seven titles altogether.
Serena Williams may be the oldest player ever to become world number one on the WTA rankings but she is still young enough to match the Grand Slam achievements of two legends, Martina Navratilova and Chris Evert.
That is the 31-year-old's ambition, says her coach Patrick Mouratoglou, now that Williams has recovered from her fitness disasters of 2010 and 2011 and has reclaimed the top spot on Monday.
After winning Wimbledon and the US Open last year, Williams' total of Grand Slam titles has risen to 15, only three fewer than Navratilova and Evert, who together are second on the all-time list behind Steffi Graf with 22.
Now Mouratoglou, who runs a tennis academy in Paris, is travelling full-time with Williams, partly in an attempt to guide her path towards these belated goals.
"Her main goal is to win as many Grand Slams as she can, as she feels that she still can," Mouratoglou said.
"That's her main motivation and I think she will keep that.
"I don't think it's hard, but she needs a few things. She needs to stay healthy first, because at 31 that's something really difficult.
"So the main thing is staying fit and the motivation. If she does those two she'll keep on improving - if she wants to. If she achieves both, I think she can still win more Slams. Why not 18? She's not that far from 18."
Some were surprised that it took Williams till now to return to the top of the rankings considering the excellence of her run in the second half of 2012.
During this she won two Grand Slam titles, the Olympics, and the WTA Championships season-end title, and captured seven titles altogether.
Despite that Williams ended last year down at number three, behind both Victoria Azarenka and Maria Sharapova.
"I don't know whether Serena cared but it was a bit shocking," Mouratoglou said.
"Nothing against Azarenka, who had an unbelievable year too, but it's a bit shocking to win two Grand Slams, the Olympics and the Masters plus some other tournaments - Serena won Madrid, Stanford, and Charleston too - and to be still number three.
"So people are saying maybe the WTA should look at how they give the points. There is a huge difference between being a finalist in a Grand Slam and a winner and maybe the points don't show how much difference. Maybe they should have a look at it.
"I understand that they want players to play all the year, but for the people who watch it's important to be able to understand tennis."
Williams says that the rankings tend to reward consistency, which Azarenka and Sharapova have both managed very well, whereas she herself has suffered from being beaten early at the Australian Open and the French Open.
"I felt I should have at least been competing for number one," Williams said.
"I played a lot of tournaments and won the majority of them. I didn't do well, though, in two Grand Slams. I think that really affected me and really hurt me in my ranking."
Williams hopes to put that right at the French Open in three months time, when she could achieve her 16th Grand Slam title.
If she succeeds and also manages successful defences of her Wimbledon and US Open titles, she will equal Navratilova's and Evert's Grand Slam totals by the end of the year.
Meanwhile Williams is in Dubai where she hopes to avenge her narrow defeat in Sunday's Qatar Open final against Victoria Azarenka, the former number one.
The Belarussian and the American are seeded to meet in Saturday's final.