The 16-year-old Norway international completed his move to the European champions from Stromsgodset on Thursday, ending months of speculation about his future after he trained with the likes of Bayern Munich, Liverpool and Arsenal.
Much is expected of Odegaard, who became the youngest player to represent Norway at the age of 15 last year and was one of the most sought-after talents in European football.
But the midfielder is confident of fulfilling his potential at the Santiago Bernabeu.
"It's a dream come true for me," he said at his official presentation. "I don't feel any pressure really. I want to become the best player possible.
"Real Madrid was the best chance for me to develop. They offered me the most opportunities."
Real confirmed that Odegaard will train with both the first team and their Castilla side.
And the youngster is happy to continue learning his trade in the reserves, but is hopeful of breaking into Carlo Ancelotti's first-team plans.
"I can't wait to start training," he added. "Whether I play in the first team of Real Madrid Castilla is not important I want to develop.
"I'd love it if Carlo Ancelotti wanted to use me."
At Madrid, Odegaard joins a squad rich in established world-class talent, led by FIFA Ballon d'Or winner Cristiano Ronaldo.
Odegaard had high praise for his new team-mates, who he has already been introduced to.
"Cristiano Ronaldo is the best player in the world," he said. "Gareth Bale and Isco are amazing players.
"I met with the Real Madrid players and they are extraordinary players."
