The Lion King (1994) is Hamlet.

"O" (2001) is Othello.

Rosencrantz & Guildenstern Are Dead (1990) is based on two minor characters of Hamlet.

She's the Man (2006) is Twelfth Night.

Romeo + Juliet (1996) is a modern-day Romeo and Juliet.

O Brother, Where Art Thou? (2000) is Homer's Odyssey. Not Shakespeare, but a brilliant modern retelling of one of humanity's oldest surviving stories. In the same vein as the above mentioned films.

These are all I can think of off the top of my head. Not to mention dozens of modern Romeo and Juliet, Hamlet, and Macbeth retellings over the years. Those three alone are the more popular Shakespeare stories for reinvention on the big screen.