"Shakespeare carried English drama to the greatest heights ever." Elaborate on this statement in the light of Shakespeare's major plays.
Now, this is one of those delightful questions where it is possible to turn the question on its head and argue the complete opposite. There are some critics who would argue that actually Shakespeare was not the greatest playwright of his age, and others, such as Christopher Marlow, produced dramatic works that were superior and more worthy of being part of the central canon of literature that we have inherited today. These critics would raise the issue that it is very difficult for us to be sure if Shakespeare was responsible himself for all of his works, and would point to the similarities in style in some of them with the plays of other playwrights such as Middleton and Marlow. This would indicate, some critics argue, that Shakespeare's plays are not actually necessarily his work entirely, and therefore that we need to be very careful about supporting such arguments as your question proposes.
However, on the other hand, if we examine Shakespeare's work and forget the arguments about ownership for one moment, we can very easily say that his literary works are incredible in the way that they have stood the test of time. This in itself points towars the way that it captures a certain universality of human experience that has enabled his plays to be translated and transported all over the globe. He is also one of the most famous Englishmen in the world, and is taught globally because of the way that he is recognised by so many as being a true genius in the plays that we wrote.
If we examine his plays, it is easy to see why. Although so many of his plays are based on earlier stories, Shakespeare has added depth of character and human emotion combined with a thematic unity that allows them to still be just as relevant today as they were in his age. Each and every play opens itself to a myriad number of possible interpretations. Just to give you one example, two weeks ago I was fortunate enough to go to the Royal Shakespeare Company theatre in Stratford, England. The production of Julius Ceasar that I saw was performed entirely by a black cast of actors, and it was set in a postcolonial African country. Such a setting really helped the audience to see the way that Shakespeare's tragedy about politics and power captures something of the timeless human condition. I have included some more links about Shakespeare and his works below, but I hope that this response will have helped you to begin to think about the statement. Good luck.