In Quebec City, the Swiss face Alexander Ovechkin and the powerful Russians, while the Czech Republic meets Sweden. In Halifax, Norway faces Canada and the United States plays Finland in the first single-game knockout round of the tournament.
Here's a look at the four quarter-finals:
Russia-Switzerland — The unbeaten Russians were all over Switzerland in the final round-robin game for both teams on Monday, taking a 4-0 lead before letting up in the third period and settling for a 5-3 win.
Swiss coach Ralph Krueger said getting a feel for the high pace at which the Russians played in that game will help his side cope with the onslaught they expect to see again.
"We can only hope that they take that confidence and that feeling of having the Ovechkins and the (Ilya) Kovalchuks coming down at you - something they never see all year - and learn how to deal with it," said Krueger.
Swiss hopes were boosted by their three third-period goals, when they opened up their attack after two periods of laying back.
"They're a skilled team in the offensive zone but they're not the greatest guys to play without the puck," said Swiss defenceman Goran Bezina. "They're not the best defensive team in the world and we can use that.
"We saw that we can score on the power play, but to create power plays we have to go into their zone and skate."
But the Russians learned a lesson of their own. For two periods, they allowed very little skating in their zone, holding Switzerland to eight shots, but then let them into the game in the third.
It is that sort of letdown that has sunk many talent-laden teams from Russia, which has not won a world championship since 1993.
Russia also has gifted skaters like Alexander Semin, Sergei Fedorov, Maxim Afinogenov and Alexander Radulov, a crowd favourite who played his junior hockey at the Pepsi Colisee for the Quebec Remparts.
A defence group that includes Andrei Markov, Fedor Tyutin and Dmitri Kalinin, backed by San Jose Sharks goalie Evgeni Nabokov, is another challenge for the Swiss shooters.
"It's great to play against those guys," said Swiss forward Julien Sprunger. "We want to show them we can play hockey.
"It will be a dream to beat a team like that with so many stars."
Only a few Swiss skaters, like Paul DiPietro and Julien Vauclair, have any NHL experience, but their best hope lies in their goaltending from NHLers Martin Gerber of the Ottawa Senators and Jonas Hiller.
The winner moves on to play the winner of the Finland-U.S. quarter-final.
Czech Republic-Sweden — Sweden is the defending Olympic champion, but is an underdog at the world championship without Senators captain Daniel Alfredsson, Mats Sundin, Peter Forsberg or Detroit Red Wings stars Nik Lidstrom, Henrik Zetterberg and Tomas Holmstrom, not to mention Johan Franzen.
But Sweden beat the Czechs 5-3 in their final round-robin game, wasting three one-goal leads before Patric Hornqvist got the game-winner with five minutes left to play.
The Swedish advantage is in goal. Henrik Lundqvist has shone while the Czechs' Milan Hnilicka has been only average.
"We've been playing better and better," said coach Bengt Gustafsson, whose attack has been led by Mattias Weinhandl, who has 10 points, and linemate Tony Martensson, who has nine. Otherwise, Sweden's best-known forward may be Washington rookie Nicklas Backstrom.
The Swedes also hope to have veteran Kenny Jonsson return from a back injury, which would add a big piece to their defence.
The Czech power play has been hot since the start of the tournament and struck twice against Sweden. Point man Tomas Kaberle leads the team in scoring with a goal and nine assists.
And there are shooters to contend with like Tomas Plekanec, Martin Erat and Patrik Elias, who banged up a knee against Sweden but is expected to dress.
"The quarter-finals make a difference between a good tournament and a bad tournament, so we have to come out strong and hard, not like we did (against Sweden)," said forward Tomas Fleischmann.
The winner will play the Canada-Norway winner in the semifinals.
Finland-United States — This one is where the action should be.
Their round-robin matchup featured more than 200 minutes in penalties, with a melee after the final whistle that saw a fight between Anssi Selmela and David Backes.
And there was controversy. Trailing 2-0 going into the third period, Finland's first goal credited to Ville Koistinen was shot through a hole in the side of Robert Esche's net. The goal stood up after video replay, prompting the IIHF to apologize and fire the video judge.
That started a three-goal third period that ended with a 3-2 win for Finland, which outshot the Americans 45-22.
The game also saw Olli Jokinen ejected for a hit from behind on Tim Gleason and a vicious hit by Dustin Brown on Jussi Jokinen. Olli Jokinen was suspended for Finland's 6-3 loss to Canada on Monday, but will be back for the quarter-finals.
The Americans, whose attack is led by NHL rookie Patrick Kane's 10 points and nine each for Brown and Phil Kessel, closed out the round robin with a 9-1 rout of Norway.
Canada-Norway — It could be a nervy game for unbeaten Canada.
In the round robin, Canada's margin of victory over lowly Norway was only 2-1 as little-known Pal Grotnes made 50 saves against Dany Heatley of the Senators, Daniel Briere, Shane Doan and company.
A Rick Nash goal with four minutes left to play gave Canada the victory.
But remember the 2006 Olympics, when Martin Gerber gave Switzerland a 49-save, 2-0 win over an even more star-studded Canadian team.
Goalies can steal games against even the best teams, although Grotnes was lit up by the Americans on a mere 48 shots on Monday.
Heatley leads the tournament with nine goals and 13 points while Nash has 10 points.
Canada will be without centre Eric Staal, who is attending his grandmother's funeral in Thunder Bay, Ont. Stall is to rejoin the team for the semifinals, should Canada advance.