Previously on Paul.
... The very familiar verse Romans 3:23 sums up this entire section: "All have sinned and are lacking the glory of God."
The second half of this verse is often mistranslated or misunderstood. The New Living Translation, for example, translates it to say, "and fallen short of God's glorious standard," as if the problem Paul has in mind is our human inability to reach God's standard of absolute perfection. But this line of thinking misses Paul's understanding of glory as something humanity lost with Adam's sin and something we are regaining and will definitively regain at the resurrection.
We can make a good argument that Psalm 8 stands behind this verse, especially the part that says God created humanity with the intent of crowning us "with glory and honor" (Ps. 8:5).  When God created humans, he put them in a position of glory and honor within the creation. But when Adam sinned, he lost that glory for humanity. When we sin like Adam did, we also illustrate why we currently lack the glory God intended for us.
Part of our restoration, whether for believers who are alive at the time of Christ's return or for those he resurrects, involves "glorification," the restoration of humanity's glory. As Hebrews puts it, Christ is "bringing many sons to glory" (2:10). Or as Paul puts it, we are going to "share in Christ's glory," a "glory that will be revealed in us" (Rom. 8:17-18). Even in the meantime, we are already "being transformed into his likeness with ever increasing glory" (2 Cor. 3:18).
Both Jews and Gentiles have lost this glory. Ever since Augustine, Christians have tended to read the "all" in "all have sinned" in individualistic terms. All individuals have sinned. Certainly Paul would agree that all individual human beings have sinned. But this is a subtle shift from what Paul's argument was really about, namely, the fact that both Gentiles and Jews had sinned. "All" in the argument meant "all groups," not just Gentiles, but Jews as well. 
When Paul says "all," he is summarizing his earlier argument. For Jews hearing Romans, the first chapter would repeatedly cue in their minds the kinds of sins that Gentiles do. They "exchanged the glory of the immortal God for images made to look like mortal man and birds and animals and reptiles" (1:23). The kind of self-righteous Jew Paul pictures in 2:17 would have "Amened" Paul at this point. "You preach it, Paul. Those sinful Gentiles are going to fry!"
Paul goes on in Romans 1:27 to speak of men who "abandoned natural relations with women and were inflamed with lust for one another." Again, sexual immorality was one of those stereotypical sins Jews assumed all Gentiles were doing. Homosexual sex was just one consummate example of the kinds of sexual sins a self-righteous Jew might point out to emphasize how sinful those Gentiles were. "Yep, preach it, Paul. Those sinful Gentiles are going to fry!"
But Paul is engaged in a sting operation. The point of Romans 2 is that Jews are just as sinful as those stereotypically sinful Gentiles. In fact, Paul knows some Gentile Christians who have the Law written on their hearts (Rom. 2:15). They put the imaginary, self-righteous Jew Paul has been picturing to shame. They have "circumcised hearts," which makes them truly circumcised in God's eyes, while the condemning Jew Paul pictures proves to be uncircumcised in heart before God (Rom. 2:28-29).
So Paul's point in Romans 1:18-3:20 is that God's judgment is coming on all ungodliness, not only on ungodly Gentiles who worship idols and are sexually immoral, but on all Jews as well. All groups have sinned--Jews as well as Gentiles--and all equally face God's wrath. This is the problem we all face. And all can also participate equally in the solution: the faith of Jesus Christ!
 We know that Psalm 8 featured in Paul's thinking about Adam because he alludes to it in 1 Corinthians 15 when he says that God is putting everything under Christ's feet (1 Cor. 15:27). Hebrews 2:5-11, which probably was written by someone associated with the Pauline mission, may give us a fuller version of Paul's thinking here. You can see my understanding of it in my sense of what stands behind Romans 3:23b above.
 This was one of Krister Stendahl's main points in his groundbreaking lecture, "Paul and the Introspective Conscience of the West."