Attacking midfielder Rodriguez becomes Real's second signing of transfer window, further boosting their creativity following the capture of Toni Kroos from Bayern Munich.
The arrival of the Colombia international comes on the back of a stunning World Cup campaign in which he led Jose Pekerman's men to the quarter-finals.
Rodriguez scored six goals to win the tournament's Golden Boot award, delivering on the promise that had been apparent ever since his days at Porto.
And the 23-year-old will now look to take his World Cup form into his maiden club campaign with Real, who will hope their transfer business brings further success after winning the UEFA Champions League for a 10th time last season.
The move sees Rodriguez depart Monaco after just one season at the Stade Louis II.
Rodriguez left Porto for Monaco for a fee in the region of €45 million in May last year.
He quickly justified that outlay, scoring 10 goals and being named in Ligue 1's team of the season as Monaco finished second behind Paris Saint-Germain.
However, the cash-rich principality club will now have to find a replacement for the playmaker as they seek to bridge the gap to defending champions PSG under new coach Leonardo Jardim.
