In the New Testament, which quotes from the Old, God says that Christians should not take vengeance on people who have wronged them:
Beloved, never avenge yourselves, but leave it to the wrath of God, for it is written, “Vengeance is mine, I will repay, says the Lord.” [Rom. 12:19; cf. Deut. 32:35]
Vengeance is something that God will take at the right time, and we should leave it to him to sort out (there’s also always the possibility that we’ll get it wrong, as well).
In fact, God wants us to have a very radical attitude towards the people who have wrong us: instead of taking vengeance, we are to show love to them. Romans 12, quoted above, goes on to say:
… never avenge yourselves … To the contrary, “if your enemy is hungry, feed him; if he is thirsty, give him something to drink; for by so doing you will heap burning coals on his head.” Do not be overcome by evil, but overcome evil with good. [Rom. 12:19-21; cf. Prov. 25:21-22]
It’s been said that if we take vengeance on people then it only perpetuates a cycle of vengeance, where vengeance leads to more retribution, which leads to further vengeance, and so on.1 God, in his wisdom, wants us to break the cycle: he wants us overcome evil by doing good to people who wrong us.
1. ‘Feud’ in Hutchinson Encyclopedia (2009)