Nearly every commentator on Romans recognizes that 1:16-17 contains the theme of the letter. Paul boasts of His gospel because in it the righteousness of God is revealed, from faith to faith.
Everyone agrees the verses are crucial. But there's little agreement about what the verses mean.
In the medieval West, the iustitia Dei was understood as God's own justice, His determination to punish evil and to reward the good.
For Luther, that wasn't good news at all. He knew he was unrighteous, and so the prospect of God "giving everyone what he's due" filled him with dread.
Luther concluded that god's righteousness isn't His own justice, nor His action in ruling justly, but the righteousness He gives, freely, by grace, to those who believe. The good news is that God justifies the ungodly.
In recent decades, N.T. Wright and others associated with the New Perspective on Paul have argued that "righteousness" is a divine attribute, not a gift. It refers to Yahweh's covenant faithfulness, His determination to fulfill His promises to Abraham and Israel, and so to restore and glorify creation, no matter what.
One way to sort through these options is to remember the content of the gospel as Paul describes it at the beginning of Romans 1: It's the good news that Jesus has been raised up as Davidic king. Whatever the "righteousness of God" is, it must have something to do with that royal announcement.
That context suggests that dikaiosune ("righteousness") is an attribute of royalty. The good news reveals God's royal justice.
That suggestion is supported by the way the Old Testament uses the phrase "righteousness/justice of God."
It refers to Yahweh's legal and judicial activity. He is just in that He issues good laws, decrees right judgments, punishes the wicked, as a good king should (Psalm 9:4-5; 119:7). That punishment of the wicked is part of what Paul means by "righteousness" is evident from the parallels between Romans 1:17 and 1:18: The righteousness of God is revealed . . . wrath from heaven is revealed.
As a good king, Yahweh also displays His justice in delivering His people from their enemies and defending them from assaults (Isaiah 51:1). The righteousness of God refers to His vindication of those who are in the right, His action in their favor (Judges 5:11).
As Wright says, the righteousness of god includes His reliability and trustworthiness. As a just King, Yahweh keeps His commitments (Daniel 9:7). His justice is manifest in His speech; He speaks what is right (Isaiah 45:19).
This royal conception of righteousness encompasses the medieval understanding (punitive justice, which is good news for those oppressed by the wicked); Luther's view (since God's justice is a saving, gracious justice); and Wright's view (justice includes covenant faithfulness).
But all these are dimensions of a more embracing concept, which we can summarize this way: Righteousness includes all the qualities, activities, and actions characteristic of a good king. The righteousness of God is His royal justice.
And in Romans 1:17, Paul is saying that the good news of Jesus manifests the righteousness of God in precisely this sense. In Jesus, God the Creator has shown Himself to be a just ruler of His cosmos.