Madness, both real and imagined, is never far from the surface in Hamlet. The lead character feigns madness as part of a plan to unnerve his wicked stepfather, Claudius, and make him feel insecure on his usurped throne. So long as Claudius is preoccupied with the seeming insanity of his stepson, he can never feel secure as king. Hamlet has turned madness into a kind of political weapon, one of the few he has in his armory. At the same time, it's much more subtle than the usual methods of gaining revenge. Hamlet's more of a thinker than a doer, and putting on an "antic disposition" and playing cat-and-mouse with Claudius is entirely in keeping with his more cerebral character.

Ophelia's madness, however, is all too tragically real. She has been driven to this sad state by her inability to reconcile the two most important roles in her life: dutiful daughter and sister, to Polonius and Laertes respectively; and Hamlet's paramour. Her cruel rejection at the hands of Hamlet ("Get thee to a nunnery!") unravels a desperate plot hatched by Polonius to figure out what's been bugging the student prince. Ophelia has been cynically used by her father as an unwitting pawn in a game of high court intrigue. Hamlet's brusque disavowal of any feelings for her leaves her with nowhere to go. Ophelia is much too delicate, too gentle for this cruel and bloody world, and her frail personality simply cannot handle the sudden strain placed upon it by all the men in her life. And so she begins a slow, but steady descent into outright madness.

Madness is an ongoing theme throughout Hamlet. It emerges primarily in two characters, Hamlet and Ophelia, but other characters comment on it, speculate about it, etc. This makes it seem like madness is a common possibility, and something that happens to people, like a storm.

In Hamlet's case, the main suggestions for why he might be mad are love for Ophelia and the loss of his father/remarriage of his mother. However, he is but pretending to be mad, so he can investigate Claudius.

In Ophelia's case, she actually loses it. She ends up dying as a result, unbalanced by the loss of Hamlet's love, his actions towards her, and, of course, his killing of her father.

What are the themes of Hamlet by Shakespeare?

One theme in the play is revenge.  It's a popular Shakespeare theme in a bunch of his plays, so it shouldn't surprise readers that he uses the theme again.  At a simple interpretation, the play is about Hamlet trying to get revenge for the murder of his father.  But a more in depth reading of the text shows the reader that the play is less about the act of revenge and more about the validity of revenge itself.  Hamlet wrestles through most of the play with whether or not to do the vengeful act.  

Lies and deceit is another theme.  It's weird that Hamlet hates deceit as much as he does, because he lies to an awful lot of people throughout the play.  

Along the lines of the lies and deceit is another theme.  The theme of madness.  Is Hamlet really crazy or is it all an elaborate deception?  

One last theme to mention.  The theme of mortality and death.  Like all Shakespeare tragedies, death is everywhere.  The play starts off with a death and ends in a bloodbath.  In between all of that, the audience sees Hamlet holding up skulls and wondering about death.