I used to think Shakespeare was overrated and didn't 'get' what the hype was about - until I read "Macbeth" at the first year at the university. That one converted me instantly. For some reason, "Romeo and Juliet" and "Hamlet" didn't do it for me when I had to read them in high school (though the fact that they were in translation might have been a part of this), and I also remember I wasn't impressed by most of the more traditional adaptations of Shakespeare's plays I had seen by that point, mostly BBC stuff... I remember also having read Charles and Mary Lamb's "Tales from Shakespeare", which, I think, did more harm than good. After I really discovered Shakespeare at the university and actually read all those plays whose plots I thought I knew, I realized that he is incredibly ambiguous, and can be interpreted in completely different ways. Once you get away from the traditional interpretations and read the text itself, they become a lot more interesting. Take, for instance, "Henry V": traditional wisdom tells you that it's a jingoistic ode to the historical English king, and two two famous film adaptations have treated it as such; but when you read the play, there are many moments that can be seen as subtle (and at least in one case, not so subtle) hints that Henry is in fact an asshole who's out to plunder another country. Of course, the play must have been written and performed as an ostensible ode to Henry, but I can't help thinking that wily Willy was able to insert some slightly subversive views, but so deftly that it would go over most people's heads. Or "The Merchant of Venice", written after the execution of Queen's physician Lopez, a Jew from Portugal who was wrongly accused of working for Spain, and no doubt drawing on the atmosphere of anti-semitism: Shylock is the villain, yet he becomes somewhat sympathetic, so that you may even seen him as a victim. We can't know what the original performances were like (after all, the plays were written to be played and never meant to be published), but it's interesting to see all the different interpretations of Shakespeare's plays during the centuries, and how they changed in accordance with the times.
It is a great irony that Will Shakespeare is now seen as an example of elite culture. He was, of course, a pulp writer, as someone mentioned before. His plays are very colorful, full of violence, sex (well, sex talk - there couldn't be any actual sex due to the obvious reasons, i.e. the circumstances of the Elizabethan theatre of the time), strong, primal emotions. He was the equivalent of a modern day screenwriter, his plays were popular with the audiences of all strata of society, and despised by university-educated rivals. But that just goes to show how silly it is to make distinctions between 'high' and 'low', elite and popular culture. He was a great poet, but I don't know if I'd say that he's unrivaled in the English language. I do, however, think that, as a playwright, and even as the author of sonnets, he has something that puts him above all others of his era, and even of the next couple of centuries. There is a depth, complexity and, as I mentioned, ambiguity to his characters and their motives. He dealt with universal human themes that can be easily applied to any time and place, and - since he was writing to entertain an audience rather than fulfill some rules of what a tragedy or comedy should be like - his play transcend genres, mix tragedy and comedy, and feel very exciting and lifelike.
I am among those who particularly enjoy adaptations of his plays in a contemporary setting - and really, I think this is much more in the spirit of the original plays, which were, as far as we know, performed in the contemporary setting of the day (e.g. everything, from "Hamlet" which is supposed to take place in 10th century Denmark, to "Julius Caesar" or "Antony and Cleopatra", was performed in 16/17th century costumes and setting). We know that historical and geographical accuracy was not Will's main concern. These plays might be ostensibly about other countries and time periods, but in essence they were meant to be about things that the viewers could relate to. It only makes sense to do the same today.
(Oh, and the 1995 film version of "Richard III" with Ian McKellen is simply awesome and should be shown to students who want to know the play.

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As for the language... well, a lot of it is not that difficult once you get the hang of some archaic expressions, and if you have notes about outdated words and words that have changed their meaning. True, some of the language is still difficult, but that's mostly in the big soliloquies (I remember that Macbeth's were quite difficult to read at first)... but if you want to see REALLY difficult language, try Milton.

Shakespeare at his most verbose and difficult is a piece of cake in comparison!
I like Shakespeare. Marc Alaimo, Jeffrey Combs and Casey Biggs did some scenes from the works of Shakespeare (with Combs playing an improbable Juliet) during their appearance at Creation Vegas. It was very well received.
They concluded their performance with a reading of this:
Are you one who considers Shakespeare's words outdated, difficult to understand?
When you hear Shakespeare do you say it's Greek to me?
You are quoting William Shakespeare.
Have you ever enjoyed a game played fast and loose?
Have you suffered from green-eyed jealousy?
Do you ever remember being tongue-tied or in a pickle? Have you ever been a laughing stock?
If the truth were known, you are quoting Shakespeare.
Have you slept not one wink?
Did you suspect foul play in an attempted assassination?Were you waiting patiently as the truth will out?
Have you stood on ceremony and insisted on fair play?
How many have stubbornly refused to budge an inch?
Do you consider you are more sinned against than sinning?
Is it your habit to act more in sorrow than in anger?
Have you knitted your brow without rhyme or reason?
For goodness sake, you are quoting Shakespeare.
Can you be a tower of strength for your family?
Will you admit when it’s too much of a good thing?
Have you seen better days?
Did you ever consider something to be an eyesore?
To give the devil his due, you are quoting Shakespeare.
Did you ever laugh yourself into stitches?
Or wish you were dead as a doornail?
Is it high time to send me packing?
I would really love to see them perform "What Shakespeare Left Behind". Combs as Juliet, LMAO.

What other plays did they perform extracts from, and how did they "cast" the characters?