Real's headline catch Rodriguez unveiled to huge crowd at Bernabeu
Colombian striker formerly at AS Monaco James Rodriguez poses during his presentation at the Santiago Bernabeu stadium following his signing with Spanish club Real Madrid in Madrid on July 22, 2014 - by Pierre-Philippe Marcou
Rodriguez, the top scorer at the World Cup in Brazil, has moved to the European champions from Monaco on a six-season contract.
Spanish media said Real paid about €80 million ($108m) for the 23-year-old playmaker, which if correct would make him the fifth most expensive player in history after Gareth Bale, Cristiano Ronaldo, Neymar and Luis Suarez.
"It's a lot of pressure to be here but I'm ready to take it on," said Colombia's poster boy, looking relaxed and smiling and sporting Real's number 10 shirt - last worn by Mesut Ozil.
"My goal here is to always win, I don't like losing," he added.
"I'm thrilled to be here, it's a dream come true, I hope to give a lot of pleasure," he told a huge crowd estimated at about 40,000, including a significant number of Colombians.
"I think Colombia will now be a fan of this great club," he added.
Neither club confirmed the size of the transfer, but Monaco described it as "one of the biggest transfers in football history".
Spanish sports daily Marca, which has close ties with Real Madrid's board, said Real will pay €75 million straight away and up to five million more in add ons.
Monaco paid €45 million ($60m) when he signed from FC Porto of Portugal.
Monaco said in a statement that it had not wanted to sell Rodriguez but that it was "proud to have carried out one of the biggest transfers in football history".
Rodriguez's six goals for his country at the World Cup won him the Golden Boot award. A spectacular volley against Uruguay went down as one of the best goals of any World Cup.
Before being unveiled he had posted on Twitter: "The dream already is REAL. Happy to be part of the best club in the world"
He is Real's second major signing in five days as they prepare for a new season of confrontation against eternal rivals Barcelona.
German midfielder Toni Kroos moved from Bayern Munich on Thursday, just days after the 24-year-old helped Germany win the World Cup.
Real have also been linked with a move for Keylor Navas, the goalkeeper whose heroics helped Costa Rica reach the World Cup quarter finals. Navas plays with Levante in the Spanish first division.
Rodriguez had only been at Monaco for one year but his sizzling performance at the World Cup made him the number one target for many European clubs.
In a FIFA.com poll released Monday, his volley against Uruguay in the last 16 was declared goal of the World Cup by more than four million voters -- three quarters of those who took part.
- A new 'special' one -
Rodriguez went to the World Cup as Colombia's second best known striker. But Radamel Falcao was injured in January leaving the starring role to his Monaco teammate as Colombia reached the last 16 for the first time.
Uruguay's coach Oscar Tabarez compared Rodriguez to Argentine legends Diego Maradona and Lionel Messi, the "players who succeed in doing things because they are special".
The Colombian has made it clear for a long time that it was the Spanish league that he wanted to get to.
"The Spanish league is a league that I've always watched -- it's different from the English League, which is very physical," he told FIFA.com in an interview in November.
"The game in Spain is more about technique, and about getting more touches on the ball. That's why I like it so much."
Rodriguez started his career at 16 for Colombian second division side Envigado FC and built up an immediate reputation.
In 2008, he was signed by Argentinian side Atletico Banfield where he became known as "James Bond of Banfield" after scoring an extraordinary left-footed lob for the unfashionable side in February 2010.
He moved to FC Porto in Portugal where more spectacular goals brought three league titles in three seasons. Monaco bought him in May 2013.