The Duchess of Malfi falls into a specific sub-genre of early modern tragedies called revenge tragedy. As one could guess, the primary attribute of revenge tragedies is that their plots revolve around the pursuit of revenge and the ensuing consequences of vengeful behavior. Thomas Kyd's The Spanish Tragedy (1587) is often credited as the first early modern revenge tragedy to have been performed on the stage. The most famous revenge tragedy from the English Renaissance is William Shakespeare's Hamlet, in which the titular main character seeks revenge on his uncle for killing his father and assuming the throne of Denmark.
Webster's play is a Jacobean revenge tragedy, meaning it was written and performed during the reign of King James I (The Spanish Tragedy and Hamlet, by contrast, were both Elizabethan revenge tragedies, performed prior to Elizabeth's death in 1603). The Duchess of Malfi represents a unique take on the revenge plot. Traditionally, avengers in revenge tragedies are also the protagonists of the play. In this case, the Duchess of Malfi is the protagonist of the play, but she is also a victim of revenge rather than the perpetrator. It is her two brothers, Ferdinand and the Cardinal, who serve as the avengers and the antagonists.
While this structure can be attributed to Webster's whims as a playwright, many consider this "inversion" in The Duchess of Malfi a commentary on the phenomenon of gender roles within the revenge tragedy genre. Put simply, women were not avengers, and they were almost always casualties of avengers' behavior (think, for example, of Ophelia's tragic death in Hamlet). The Duchess of Malfi maintains this gendered structure despite its portrayal of the Duchess as clever, strong, and determined. In the end, she is a victim of her brothers' desire for revenge, a tragic conclusion that emphasizes the play's pessimistic outlook on humanity.