Coventry City beat Wolves in their quarter-final thanks to Haji Wright's 100th-minute winner.
Manchester United turned things around in extra time to beat rivals Liverpool at Old Trafford.
Meanwhile Manchester City cruised past Newcastle in the last eight and Chelsea beat Championship leaders Leicester.
Both semi-finals will take place at Wembley Stadium on the weekend of 20-21 April.
Coventry have not played in an FA Cup semi-final since they won the competition back in 1987, the club's only major trophy.
They are managed by former Manchester United striker Mark Robins, who helped the Red Devils win the FA Cup in 1990.
Twelve-time winners Manchester United last lifted the FA Cup in 2016, and were runners-up in the competition last season.
Holders Manchester City are looking to reach their 13th FA Cup final and are chasing an unprecedented second consecutive Treble.
Chelsea have reached five of the last seven FA Cup finals, last winning the competition in 2018.
-- Courtesy of BBC Sport