Beowulf exists as a text of the Anglo-Saxon period. Readers only see one distinct type of revenge in the epic poem: destruction of sippes/comitatus (sippes: one's clan, kin, or extended family; comitatus: the friendship between a king and his men). This revenge is seen twice. First, after Beowulf kills Grendel, his mother goes to Heorot to enact revenge (destruction of sippes). Then, Grendel's mother murders Aeschere (Hrothgar's counselor and friend). In an example of destruction of comitatus, Beowulf then, in order to enact revenge upon Grendel's mother for the murder of Aeschere, murders Grendel's mother.
Outside of destruction of sippes, Anglo-Saxons sought revenge based upon feuding and betrayal. Feuding and betrayal (in any sort or sense) were both found to be grounded reasons for enacting revenge.
One way to combat, or end, revenge in the Anglo-Saxon culture was to pay a wergild (the value of a person's life or piece of property). The Anglo-Saxons defined a value based upon the ranking of all people within the society. If murdered, the charged party would be required to pay the value of the person's life to his or her family.