Her first Grand Slam title: Serena beat Hingis, 6-3, 7-6(4), to win the 1999 US Open:
What happened: At the moment of victory, Williams wasn't sure whether to "laugh, cry, scream or jump", so she ended up doing all four. The youngest of the Williams sisters was the first of the pair to score a major, and no one can say that the 17-year-old won a 'soft' title. On the way to the final, Williams had defeated the likes of Kim Clijsters, Conchita Martínez, Monica Seles and Lindsay Davenport, and then in the all-teenage title match she had to fend off an opponent who had already experienced winning Grand Slams. At 5-3 in the second set, Williams had a couple of championship points, which she didn't convert, and suddenly Hingis was playing her best tennis of the match. But a voice in Williams' head was telling her: "Serena, this can't happen, you have to think positive things." Williams listened to that voice.
What Serena said: "Oh, my, God, oh, my, God."
What her opponent said: "You can't push the ball back at Serena - she just kills it. It was terrific tennis."
What others said: "This is my proudest moment - words cannot express how I'm feeling," Williams' father, Richard, disclosed.
The statistic of the match: Williams became only the second African-American woman to win a Grand Slam title, the other being Althea Gibson, whose last major had come 41 years earlier, in 1958.
A memorable moment: Minutes after Williams left the court, she took a call from Bill Clinton and his daughter Chelsea. "I thought for sure my day couldn't get any better, and the next thing I knew someone was telling me, 'The President of the United States wants to talk'. I was thinking, 'Wow'."
Her second Grand Slam title: Serena beat Venus, 7-5, 6-3, to win the 2002 French Open:
What happened: Two and a half years had passed since Serena's first major title; this was her showing, with victory over Venus, that she wasn't "a one-hit wonder". For the first time, the sisters, housemates and dearest friends, met in the final of a Grand Slam tournament, and both seemed tense and on edge, and the result was a match of an uneven quality. "I think that both of us wanted to win so much that we were a little tight out there," Serena said. "There were so many ups and downs, especially in the first set. We kept breaking each other and at some point, I really thought, 'Oh, gosh, our father is not going to be happy to see us play like that.'" This victory in Paris would prove to be the first leg of what became known as the 'Serena Slam' - the younger sister would go on to hold all four majors simultaneously.
What Serena said: "I want to thank Venus for supporting me the whole way and for being the best sister in the whole world."
What her opponent said: "I'm trying not to be too sad about my loss. I'm happy for Serena because she hadn't won a Slam in a while."
What others said: "Serena wanted it more - it had been too long since her last Grand Slam title," said the sisters' mother, Oracene.
The statistic of the match: It was only the third time in tennis history that two sisters had contested a Grand Slam final, the previous two occasions being when Maud and Lilian Watson met for the 1884 Wimbledon title, and when Venus and Serena had played at the 2001 US Open.
A memorable moment: After the prizes had been awarded, Venus joined the press photographers to take her own images of Serena holding La Coupe Suzanne Lenglen.
Her third Grand Slam title: Serena beat Venus, 7-6(4), 6-3, to win 2002 Wimbledon:
What happened: Driven by a desire to "become a part of history" at the All England Club, there was a ferocity to Serena's tennis on Centre Court. For the second successive Grand Slam, Serena beat her older sister Venus, but this grass court match was very different to their final in Paris a month earlier. Listening to her great roars of "C'mon", it was plain how badly Serena wanted to attend the Champions' Dinner. So Venus had been hoping to win what would have been a third successive Wimbledon title. But the defending champion had a sore shoulder that prevented her from serving at full speed, "a weakness" that her sister exploited. "Obviously, I'm a competitor, and unfortunately, it's war out there," Serena said. "If there's a weakness out there, someone's going to have to be attacked."
What Serena said: "At the beginning of the year, I told myself that I didn't care what else happens this year, I wanted to win Wimbledon. I really wanted to win the French Open but that was just an extra bonus. I got a little lackadaisical after winning the first set, a little too satisfied, and I hit a couple of ridiculous shots. I thought to myself, 'Some day you're going to be telling your grandchildren about how you blew it'. So that's when I decided I would go ahead and take my opportunity. I kept thinking to myself, 'OK, Serena, you stay calm. Venus already has two Wimbledon titles. Try to fight.'"
What her opponent said: "Serena played a better shot every time, and sometimes you have to accept that. It's great to see Serena doing well because for a while there she wasn't doing her best. Now I think she has to feel better that she's taking full advantage of her career."
What others said: "They put blood aside. Something few thought they could ever do," observed The New York Times. "The proof was not just in the quality of the match, but in the attack mode Serena was in."
A memorable moment: The cameras showed that Serena spent the changeovers reading - she had brought on notes reminding her to "stay low" and "hit in front of you".
Her fourth Grand Slam title: Serena beat Venus, 6-4, 6-3, to win the 2002 US Open:
What happened: For all the attention at the time on what Williams wore at Flushing Meadows - a black catsuit - it is her tennis that you remember. Once again, it was the younger of the sisters who played with greater authority and class during a Williams versus Williams Grand Slam final. Their father and coach, Richard, became so tense during the second set that he walked away: "I'm as nervous as hell - I'm getting out of here." All those who stayed in the Arthur Ashe Stadium knew how this was gong to end, with Serena getting the better of her sister for the third successive major. Having missed the Australian Open because of injury, Serena was undefeated at the majors all season, and three quarters of the way to her non-calendar year Grand Slam, otherwise known as the 'Serena Slam'.
What Serena said: "I'm just happy to win here again. It's been a long time since 1999. I had almost forgotten how I felt. The difference between then and now is that I'm a bit more mature, more relaxed and a better player. I just have more fun with what I do and I'm not as stressed out there as I used to be."
What her opponent said: "Everyone has their year and this has been Serena's year."
What others said: "Serena played like a No.1. Venus, as usual, suffered with her serve and with everything else," said their mother Oracene.
The statistic of the match: This was only the second "primetime" final in the history of the tournament, the first having been the previous year when Venus had defeated Serena.
A memorable moment: When the Star-Spangled Banner was being played after the match, there was a show of the great affection the sisters have for each other, with Serena whispering something in Venus's ear, though neither of them would share what the champion had said. That would stay "secret", Serena said.
Her fifth Grand Slam title: Serena beat Venus, 7-6(4), 3-6, 6-4 to win the 2003 Australian Open:
What happened: The Serena Slam was completed on Melbourne's Rod Laver Arena. A first Australian Open title for Williams made her only the fifth woman in tennis history - after Maureen Connolly, Margaret Court, Martina Navratilova and Steffi Graf - to hold all four majors simultaneously. It also gave her a Career Grand Slam. In the semifinals, Williams had come close to defeat - she had trailed 1-5 in the third set against Kim Clijsters, and had also faced a couple of match points - but she didn't live so dangerously when playing for the title against her sibling. All four of the majors that went into Serena's Slam had come through defeating Venus, who became the first player in history to lose four successive Grand Slam finals.
What Williams said: "I can't believe I can be compared to these women who also held all four Grand Slams simultaneously. To be in their company is amazing. I am making history now, and it doesn't happen every day. It's been tough. All my life, I've dreamed of being the best, and doing the best, and it hasn't always been easy for me."
What her opponent said: "Not many players get all four Grand Slams at the same time, so this is really wonderful, it really is. But I don't like losing altogether. Whether it's against Serena or against any other player, it's never fun."
What others said: So emotional was Williams in victory that the occasion brought "the rare sight of a choked-up Serena," according to The Melbourne Age.
The statistic of the match: It was 45 degrees Celsius in the city that day, which meant that the Australian Open invoked their 'Extreme Heat Policy', closing the roof over the Rod Laver Arena, and making this an indoor Grand Slam final.
A memorable moment: The sisters said they were playing for the memory of their late grandmother, Ora Lee Price.
Mark Hodgkinson is the author of 'Game, Set and Match: Secret Weapons of the World's Top Tennis Players' (Bloomsbury, May 2015).