"Vengeance is mine, I will repay, says the Lord"
In the second half of Romans 12, Paul is calling on us in view of God's mercies to love others, and especially return evil done to us with good. To help us repay evil with good, Paul reminds us of the good news of God's vengeance.
How is God's vengeance good news? It is good news because we are reminded that God will right every wrong. When we are tempted to repay evil for evil, Paul reminds us of God’s vengeance. This reminder works two ways. If we repay evil with evil, we are doing injustice against another and we are sinning against God. It works another way, too. We don't have to take out vengeance on another, because God will do it. We are to leave room for the wrath of God and not take vengeance. What we are saying, when we do take vengeance, is God can't be trusted to right this; I have to take it into my own hands.
Ever notice that in action movies God is missing from the worldview? Why? Because the hero of the movie is the action hero who “brings justice.” Notice that the justice always ends with the antagonist receiving death at the hands of the protagonist? In a worldview without a God of justice, it's up to us to do the justice.
But in the real world, one with a just God at the center, we can let the authorities punish wrong doing (Romans 13.4), as God's servant. But even if the wrong doing is not properly punished by the state, God will bring it to rights, so we must not retaliate evil for evil but overcome evil with good. Notice that taking vengeance out on another is not only responding with violence to violence, but can also be as simple as giving the silent treatment to another when wronged. The silent treatment is repaying evil for evil. What is the silent treatment? It is punishing another for their evil to us. Instead, Paul says to do good to others, even in the small day to day evils done to us.
Vengeance is good news as well—because not only does God remind us that he will right all wrongs—it is good news because God's wrath against our sin has been dealt with in Jesus' life, death, and resurrection. So, if we do good to another sinning against us and they don't repent, God will have vengeance on them—so we are not to respond to evil with evil. If the sin committed against us is committed by a fellow disciple of Jesus, that sin is paid for by Jesus' death, just like all of ours are. So, if we are forgiven much sin, and they are forgiven much sin, how can we justify any form of vengeance against another disciple of Jesus?