Federer's win sets up an intriguing encounter against Davydenko, who has beaten the world number one in their previous two encounters.
BBC: Roger Federer eases into Australian Open quarter-finals
Kvitova's win sets up a match against another French opponent, Marion Bartoli.
The win sets up a hugely compelling Masters next month with Woods finding his game, the new kid on the block in McIlroy and four-time Masters champ Phil Mickelson playing well.
The win sets up an All-Ireland semi-final meeting with Connacht champions Corofin in February and on this evidence, it will take a very good team to beat St Gall's in the spring.
The win sets up the reigning champions for the visit of Bayern Munich to the Westfalenstadion on Wednesday in a fixture which will go a long way to deciding the destination of this year's title.
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Serena Williams is already into the second round with a straight sets win over Varvara Lepchenko.
Argentina's Monaco registered a straight-sets win over Lukas Rosol of the Czech Republic.
Four-time grand slam champion Clijsters felt there was still room for improvement in her game despite recording a third successive straight-sets win.
The straight sets win came on the British no 1's least favourite surface of clay, and followed fellow Briton Greg Rusedksi's defeat on Monday.
Many have tipped Croatian 10th seed Marin Cilic as a dark horse for the title and he enjoyed a straight-sets win against Spain's Daniel Gimeno-Traver.
Henin is also on the comeback trail, inspired by Clijsters' exploits, and returned with a superb straight sets win over second seed Nadia Petrova on Monday.
May-Treanor and Jennings, who until last week hadn't lost an Olympic set for more than eight years, hung on to win in straight sets, but just barely.
Murray will meet Tomas Berdych in the fourth round on Sunday after the Czech Republic 15th seed enjoyed a straight-sets win over American John Isner, ranked 17.
Radek Stepanek celebrated his first call-up to the Czech Republic's Davis Cup team with a straight sets win over compatriot Jiri Novak at the Milan Indoors event.
Meanwhile, on the WTA Tour, China's Li Na reached the quarterfinals of the Stuttgart tournament with a straight sets win in her first match of the clay court season.
Roddick, who been troubled by a shoulder injury this season, looked on his way to a straight-sets win but Mahut fought back to level the match in a tie-break.
Sixth seed Juan Martin del Potro eased past German Benjamin Becker 6-2 6-4 6-2, while French seventh seed Jo-Wilfried Tsonga battled to a straight-sets win over Japan's Go Soeda.
Jetlagged on his arrival from Abu Dhabi, he suffered a scare when a crowd barrier collapsed on him during his opening match, a straight sets win over Seppi.
On the second of a pair of match points in the tie-break the line judge called Federer's drive in, a decision which would have given him a straight sets win.
Sixth seed Venus Williams, who is chasing her first Australian Open title at the age of 29, claimed her second consecutive straight-sets win as she was similarly untroubled by Sybille Bammer 6-2 7-5.
The Czech player won through to his first Masters Series final with a straight sets win over Fabrice Santoro on Saturday but said he had picked up the injury towards the end of that match.
It capped a big week for Kvitova, who beat former world No. 1 and 2011 Dubai champion Caroline Wozniacki in the semifinals after ending the title defense of fourth-ranked Agnieszka Radwanska following a straight-sets win over Ana Ivanovic.
The opening day of the second grand slam of the season did not feature the leading attractions, but French hope and fifth seed Jo-Wilfried Tsonga gave the home crowd some cheer with a four sets win over Russian qualifier Andrey Kuznetsov.
Home hopes were boosted by an easy straight sets win for seventh seed Andy Roddick over Thiemo de Bakker of the Netherlands while in-form Jo-Wilfried Tsonga of France was taken to three sets before beating Albert Montanes of Spain, 4-6 6-3 6-3.
The result won't always conform with a sense of how close the matches were: A winner of a five-setter in which he dropped two sets in tiebreakers and won the other three sets 6-0 probably would have a higher dominance ratio than a straight-sets win in which each set was decided by a tiebreaker.
Venus, the fourth seed at the clay court tournament, was broken early and lost the opener to the 15th-seeded Italian, but recovered to win the next two sets easily for a 3-6 6-1 6-2 win.
The top seed also beat Willstrop in straight sets to win the Commonwealth Games title in Delhi last October.
"He had so many chances on my serve he probably deserved to win the first two sets, " said Murray.
'The Viking' held his nerve in a thrilling match, coming back from 3-0 and 4-2 down in sets to win.
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