In The Tragedy of Macbeth, by William Shakespeare, Macbeth orders Fleance's death in an effort to "tie up loose ends" and prevent any potential threats to his reign.
From the start of the play, Macbeth and Banquo are very close. They fight together under Duncan, and they are both known for their courage and valor in battle. However, once Macbeth is given the Weird Sisters' prophecy that he "shalt be king hereafter" (1.3 33), he takes steps to not only become the King of Scotland, but also to eliminate any potential threat to his reign. Banquo, Fleance's father, was present at the time of the prophecy, and he later begins to suspect Macbeth of foul play in the death of Duncan (rightfully so, as Macbeth murdered Duncan). Further, during the initial encounter with the Weird Sisters, Banquo received the prophecy that "Thou shalt get kings, though thou be none" (1.3 70). In other words, Banquo will "father" kings. A member of his line will eventually be the king of Scotland. Just as Banquo eventually suspects Macbeth of foul play in the death of Duncan, Macbeth, as his paranoia grows, begins to believe that Banquo will plot against Macbeth's reign in part to ensure Banquo's own line gains the crown. In Act III, Scene II, Macbeth admits his fears to Lady Macbeth, saying "O, full of scorpions is my mind, dear wife!/ Thou know'st that Banquo and his Fleance lives" (3.2 41-42). However, he goes on to tell her that "There's comfort yet; they are assailable" (3.2 44). Unbeknownst to her, he has already enlisted murderers to kill Banquo and Fleance while they are out on an afternoon ride. His desire to kill Fleance stems from both his desire to end Banquo's line and to prevent any potential act of vengeance by Fleance, after his father has been murdered. This is essentially Macbeth's attempt to silence two potential threats with one act. Finally, from a dramatic standpoint, this act is the first time that Macbeth orders the murder of a child. Early in the play he struggled against his conscience before and after murdering Duncan, an aged king; the ease with which he can now order the death of a child further demonstrates just how far he has fallen.