I believe that Shakespeare, being the literary genius that he was, saw the lack of a fantasy genre and therefore created one when he wrote A Midsummer's Night Dream. By mixing local folklore with traditional dramatic structure in writing his play, the fantasy genre were born. A Midsummer's Night Dream sets standards that many modern fantasy novels embrace and expand upon.
In Shakespeare's play, Bottom (a human) is taken by the queen fairy Titania (but only after Puck's interference). This parallel universe is one where the fairies can see and interact with the humans. The humans are a very important part of their life even though they are oblivious to the fairies. While Bottom can transcend into the fairyland, he belongs in the mortal, human realm and ultimately returns to it.
In the fantasy series of Harry Potter by J. K. Rowling, most can find many similarities to the fairy world Shakespeare created but also significant differences.
Like in the play, the parallel world in Harry Potter is interwoven with the real world yet the creatures that live in this alternate universe can be seen (mistaken as humans). The main character, Harry Potter (of course), like Bottom, is introduced into this magical world by one of it's inhabitants; however, unlike Bottom in Shakespeare's play, Harry Potter actually fits in and belongs with this parallel reality.
C. S. Lewis' Chronicles of Narnia is another good example of modern fantasy that can be traced back to A Midsummer's Night Dream. In this series a group of children are taken to this other world, not by an inhabitant, but by a wardrobe. And while this parallel universe is completely detached from the real one, some of the inhabitants know of the earthly world. Like Bottom, these children are...