Victory over Australia in Oita last weekend earned England their place in the last four, while the All Blacks got the better of Ireland.
Here, the PA news agency takes a look at each team's progress through the tournament.
Pool C: England 35 Tonga 3
Manu Tuilagi touched down twice as England opened with a victory in Sapporo, but they had to wait until the 77th minute to pick up the bonus point, with Luke Cowan-Dickie crossing for their fourth and final try.
Key stat: England conceded three points, their joint-fewest in a Rugby World Cup match. They also conceded three against Romania in 2011, as well as versus Uruguay in 2015.
Pool C: England 45 United States 7
Eddie Jones' men came close to recording their first World Cup whitewash with a seven-try performance against the US, but Bryce Campbell scored for the Eagles after the final gong, with the Americans having been reduced to 14 men thanks to John Quill's red card for a shoulder to the head of Owen Farrell.
Key stat: This was the first time since 1995 that England had kept their opponents scoreless in the first half of a World Cup match.
Pool C: England 39 Argentina 10
The Pumas had Tomas Lavanini sent off for another shoulder to Farrell's head inside the opening 20 minutes, making England's job easier in what should have been their toughest test of the World Cup to that point. The 2003 champions' final Pool C clash with France was cancelled due to Typhoon Hagibis, which saw them top the group by two points from Les Bleus.
Key stat: England became the first team to claim a place in the quarter-finals with this victory.
Quarter-final: England 40 Australia 16
Jonny May crossed for two of England's four tries as he collected his 50th cap, with Jones' side racking up the most points the Wallabies had ever conceded in a World Cup fixture. The win in Oita set up this weekend's semi-final clash with the All Blacks, while the result prompted Wallabies coach Michael Cheika to stand down.
Key stat: Owen Farrell's 20 points were the most by an England captain in a World Cup match, beating the previous record of 17 by Rob Andrew against Italy in 1995.
Pool B: New Zealand 23 South Africa 13
The All Blacks opened with a clash between two titans of world rugby, which could yet be repeated in the final. Though the match never quite lived up to the hype, two tries in three first-half minutes from George Bridge and Scott Barrett gave the All Blacks a platform upon which to build a victory that got their campaign off to the ideal start.
Key stat: Barrett scored his fourth Test try and his third against South Africa.
Pool B: New Zealand 63 Canada 0
Steve Hansen's men put Canada to the sword as they won to nil for only the second time in their World Cup history, having last achieved the feat in a 40-0 win against Scotland in 2007. Scoring nine tries in total, the All Blacks had a bonus point in the bag by half-time.
Key stat: All three Barrett brothers touched down for the All Blacks, marking the first time that three siblings have all scored points at the same World Cup.
Pool B: New Zealand 71 Namibia 9
It was another comfortable outing for the defending champions against Namibia, as they chalked up 11 tries through eight different scorers. Sevu Reece, Anton Lienert-Brown and Ben Smith all touched down twice. The All Blacks' final pool fixture against Italy was also not played due to the weather.
Key stat: Sam Whitelock picked up his 17th consecutive World Cup win, surpassing the previous record of 16 held by former All Black Keven Mealamu.
Quarter-final: New Zealand 46 Ireland 14
This was a victory that made the All Blacks' challengers for the World Cup crown sit up and take notice, as they cantered to a convincing victory over an Ireland side who had beaten them twice in recent years. Aaron Smith touched down twice, with New Zealand running in seven tries over the course of the match.
Key stat: This was the third biggest winning margin for a team in the knock-out stages of a World Cup, trailing only wins for the All Blacks against France in 2015 (62-13) and Wales in 1987 (49-6).