Opinions on this one vary (and wildly, at that), but I think Julie Taymor's adaptation of Titus Andronicus is one of the most audacious and interesting Shakespeare adaptations ever made. It's a film that is alternately brutal and beautiful, borrowing liberally from various eras of history so that it's not really fixed in any one time period. It's a deeply weird movie, but that's part of what makes it so good.
To date, the only Shakespeare adaptation to be named Best Picture. Hamlet has been adapted many times, but Laurence Olivier's atmospheric, noir-tinged take is the most noteworthy. It's not just an exquisitely acted film, it's a visually fascinating one as well.
A great film from one of the greatest filmmakers of all time. Akira Kurosawa's adaptation of King Lear makes some significant changes from the source material, which it uses in conjunction with folklore, but it is an absolutely inspired film and, amazingly, the first and only film to earn an Oscar nomination for Kurosawa.
Kenneth Branagh is no stranger to bringing Shakespeare to film (his adaptations of Much Ado About Nothing and Hamlet are also deserving of consideration in a "best of Shakespeare" list), but his mastery of the material was apparent right from the beginning. His directorial debut Henry V is not just a fantastic film, it also netted him two Oscar nominations, one as Best Director, the other as Best Actor.
Richard Loncraine's adaptation, which moves the action to a Fascist Britain, is a mesmerizing triumph of a film. Starring Ian McKellan in the lead role and delivering one of his very best (and yet, shockingly unheralded) performances, and featuring a score of wonderful actors filling out the supporting ranks - Annette Bening, Robert Downey Jr., Maggie Smith, Kristen Scott Thomas and Jim Broadbent - this adaptation can't be beat.