I think this is the best filming of a Shakespeare play, in terms of overall success. The filming is straightforward, with a minimum of distractions, cuts were made to the script to keep things moving, the dialog is clearly spoken, and the performances are terrific all around.
As just about every other comment here notes, if you only know Brando from The Godfather and some of his later, and sorrier films, you will be amazed and impressed by his Marc Antony. This is the Brando that I remember, buff, gorgeous and so talented that we were sure he could play just about any part and blow us away. His performance of the famous "Friends, Romans, countrymen" speech is a marvel of clarity, and is the linchpin that makes all of the other action of the play make sense.
James Mason is, I think, perfect as Brutus. He is very much like Shakespeare's Hamlet - mulling over every possible facet of every problem he faces, and agonizing to reach a decision. He was a master at portraying a person's ability, or inability, to reach a painful decision. The awesomeness of his responsibility and the consequences of his actions (after all, they are plotting to kill a king) are beautifully shown in his performance.
John Gielgud is my favorite Shakepearean actor. If you had ever had the privilege of seeing him on stage, you would have gotten the full force of his ability to control the character, the language, and to reach out and hold the audience all at the same time. It doesn't quite come across in this film, but I still think he shows that underneath Cassius' treason there is definitely an element of self-doubt and possibly shame at what he is about to do.
I have to disagree with most of the comments about Louis Calhern's Caesar. Several people have said that he didn't capture the majesty and military bearing that Julius Caesar would have had, but we have to remember that Shakespeare intended this as drama, not history. The whole point of the Roman senators' wish to get rid of Caesar is that he is no longer the Caesar they remember: he has become a smug, self-satisfied politician who thinks he is a king, while Rome is still a republic. I think Calhern captures this smarmy, oily, arrogant quality very well. Rome wanted a general, and this Caesar gave them a high-priced car salesman.
I own a copy of this film, and I watch it often. I think it would serve perfectly as an introduction to Shakespeare. By the way, I remember an anecdote related in the memoirs of John Houseman (the producer of this film). He said someone of importance in British theater (I now forget who - possibly it was Geilgud) had observed Brando's performance in the making of the film, and asked him to come to London to star in a Shakespeare festival. Brando said sorry, I can't. I have to get back to Nebraska to help my father get the crop in. Imagine if he had said yes.
