France's bid to retain the World Cup continues against Poland on Sunday, while England's pursuit of a first major trophy in 56 years will be tested by African champions Senegal.
Didier Deschamps' French side are aiming to become the first team to win successive World Cups since Brazil in 1958 and 1962 after winning Group D in Qatar.
Led by electric Paris Saint-Germain forward Kylian Mbappe, the French are heavy favourites to progress past Poland at Doha's Al Thumama Stadium.
But in a World Cup packed with shocks like the group-stage exits of Germany and Belgium, subduing Barcelona striker Robert Lewandowski is the key to France's hopes of avoiding another upset.
Poland's all-time leading scorer will be hoping to add to his meagre World Cup tally of one goal against France goalkeeper Hugo Lloris, who is set to make his 142nd international appearance and equal Lilian Thuram as Les Bleus' most-capped player.
France captain Lloris praised both Lewandowski and his Polish counterpart Wojciech Szczesny, who has starred with two superb penalty saves which were crucial to his team finishing second in Group C.
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"Lewandowski has been one of the best number nines in the world for years now but they are a good side collectively," said Lloris.
"Szczesny is a having a fantastic tournament, Poland owe their place in the last 16 to their goalkeeper."
If France see off Poland, their path towards the World Cup final on December 18 could include a quarter-final showdown against England.
Gareth Southgate's team are slated to meet France in the last eight if they beat Senegal at Al Bayt Stadium.
England reached the semi-finals four years ago and finished runners-up at Euro 2020, and got off to a fine start in Qatar by winning Group B.
And they believe they can win the tournament in Qatar to finally end their agonising wait for a first major trophy since the 1966 World Cup.
"That is not being over-confident, that is being humble enough to see what goes on in our training and how the staff prepare, and believing in ourselves too," said England defender John Stones.
Rocked by the loss to injury of talisman Sadio Mane before the tournament, Senegal reached the World Cup knockout stages for only the second time after Tuesday's 2-1 win against Ecuador secured the runners-up spot in Group A.
On Saturday Lionel Messi helped line up a quarter-final clash with the Netherlands, comfortable 3-1 winners over the USA, with a goal and a classy performance in Argentina's unexpectedly nervy 2-1 win over Australia.
Messi got Argentina rolling on an emotional evening in front of massed ranks of boisterous fans with his 789th goal in his 1,000th career appearance, his first ever in a World Cup knockout tie.
"I know the effort they are making to be here with us and I know how much they enjoyed it," said Messi of his team's passionate supporters.
"There is a bond we have -- it is something beautiful, it is what the national team should be."
Messi opened the scoring in the 35th minute of an absorbing encounter at the Ahmad bin Ali Stadium.
His strike appeared to put the Argentines on the way to a comfortable win, particularly after Julian Alvarez pounced on a horrendous error from Australia goalkeeper Mathew Ryan to double his team's lead just before the hour mark.
However Craig Goodwin's wayward shot somehow smacked off Enzo Fernandez's face and into the Argentina net with 13 minutes remaining, setting up a heart-pumping end-to-end finish in which both teams should probably have scored.
Lautaro Martinez missed two great chances to seal the win for La Albiceleste, while Aziz Behich was denied an improbable leveller late on when after a winding run from left-back Lisandro Martinez made a vital last-ditch challenge to stop the Dundee United man just as he was about to shoot.
Possibly Australia's best chance fell to teenager Garang Kuol deep in injury time, but goalkeeper Emiliano Martinez made a desperate save to secure Argentina's passage to the next round.