Identify three instances of revenge in Hamlet, and describe them. Who is seeking revenge in each instance? What is the outcome in each incident?

One instance of revenge is with Fortinbras.  He wants revenge because his father lost a fight to King Hamlet and thus lost his territories (Act. 1, sc. 2).  Fortinbras puts together a group of mercenaries to achieve his goal.  By the end of the play, Act 5, sc. 2, Fortinbras has successfully invaded Denmark and claimed the throne.  He has achieved his goal of getting revenge.  The loser is Denmark in that a Dane no longer sits on the throne of that country.  Another instance of revenge is with Laertes.  He wants vengeance for his father's death.  Hamlet killed Polonius when Polonius was hiding behind the arras in Gertrude's chamber when Hamlet confronted her after the play (Act 3, sc. 4). In Act 4, sc. 7, Laertes and Claudius plot to have Laertes kill Hamlet when the two of them engage in a "friendly" duel.  In Act 5, sc. 2, during that duel, Laertes does scratch Hamlet with his sword, the blade of which he has annointed with poison.  During this scene, Gertrude falls having drunk from the cup of poisoned wine that was intended for Hamlet and then Laertes, realizing that Claudius was behind everything, tells Hamlet the truth.  Hamlet does die, but not before Laertes, so Laertes never does get his revenge against Hamlet.  Since Hamlet kills Claudius, both Hamlet and Laertes do get revenge against him for all the treacheries he committed including killing Hamlet's father, King Hamlet. Nearly everyone loses and dies here.