كتب : Omar AbdelAziz
Pellegrini left his post at Villarreal to take over from Juande Ramos, who was appointed on a temporary basis after the sacking of Bernd Schuster last year.
It was reported that Real have paid their Liga rivals €4million in compensation for the services of the Chilean.
Pellegrini led Villarreal to the Champions League quarter-finals this term, after they had finished second in the Primera Liga last season.
The 55-year-old joins 'the project' of new chairman Florentino Perez, who also vowed to bring in a number of high-profile signings this summer.