This will be the fourth successive season the Madrid rivals have met in the competition and Atleti have yet to get the better of Los Blancos across that run.
Real won both the 2014 and 2016 finals against their foes from across Madrid, with Diego Simeone's men now having the opportunity to exact revenge when the sides meet in the last four.
The semi-final first legs will take place on Tuesday 2 and Wednesday 3 May, with the returns the following week.
Atleti have finished runners-up in the competition on three occasions, whereas Real's victory last year was their 11th in the competition.
Gareth Bale would love nothing more than to win the competition in his hometown of Cardiff, while Juve goalkeeper Gianluigi Buffon will hope to add the trophy that has so far eluded him during his glittering career.
The 39-year-old's side will be home for the second leg against Monaco, whose young, exciting team knocked out Borussia Dortmund in the quarter-finals.