The 25-year-old has signed a five-year contract worth 15 million euros (£12.8m) post-tax after telling PSG he would be departing the club at the end of the season.
Mbappe will also receive a signing-on bonus of 150 million euros (£128m), which will be paid in instalments over five years.
The move will be made official at the start of July after the expiry of his PSG contract and he will be unveiled before jetting off for Euro 2024 duty with France in Germany.
"Real Madrid CF and Kylian Mbappe have reached an agreement whereby he will be a Real Madrid player for the next five seasons," Real said in a one-line statement.
Mbappe then posted pictures of himself on social media as a child wearing a Real Madrid, including meeting Cristiano Ronaldo.
"A dream come true," he said. "So happy and proud to join the club of my dreams.
"Nobody can understand how excited I am right now. Can't wait to see you, Madridistas, and thanks for your unbelievable support. Hala Madrid!"
Mbappe's move to the Bernabeu comes just two days after Real lifted their 15th Champions League title in a double-winning campaign after also winning La Liga.
He has been a long-standing target for Los Blanos, who made an approach to sign him in 2022 before he extended his PSG contract until 2025.
But he did not take up the option to extend for an additional year, allowing him to walk away from the Parc des Princes for nothing.
Now, he joins a star-studded squad featuring Vinicius Jr, Jude Bellingham and Rodrygo under the tutelage of veteran coach Carlo Ancelotti.
Mbappe, who joined PSG from Monaco for 180 million euros (£165.7m) in 2017, scored a record 256 goals in 308 games across six seasons.
He leaves Paris having won 15 trophies including six Ligue 1 titles but without the Champions League trophy, having come close as a runner-up in 2020.