The theme of chapter three is the availability of a righteousness from God

In the first section we have God's judgment being defended:

What advantage then hath the Jew? or what profit is there of circumcision? [Rom. 3:1].

"Profit" means that which is surplus, that which is excess, and the Question has to do with the outward badge of God’s special covenant with the Jews, circumcision.
It looks as if Paul is in danger of erasing a distinction which God has made. The question is, if Jew and Gentile are on the same footing before God, what then is the supposed advantage of the Jew and what good is circumcision? Paul is saying, "Yes, the Jew has an advantage." The advantage, however, created a responsibility. We need to note carefully the advantage the Jew had because there is a great deal of confusion in this area.

Paul is making it clear that God not only gave to the nation Israel the oracles of God—they were the ones who communicated the Word of God—but in the Word of God was something special for them, promises, not yet fulfilled, and peculiarly their own.

The Jew failed; doesn’t that mean God failed? No. God’s promise to send Israel the Redeemer was not defeated by their willful disobedience and rejection. All His promises for the future of the nation will be fulfilled to His glory in spite of their unbelief. I personally thank God that His promises to me do not depend on my faithfulness. If it had depended on me, I would have been lost long ago.

Now, the whole point is this: if my unrighteousness reveals the marvelous, wonderfully infinite faithfulness of God in the grace of God, then has God a right to judge me? That’s what Paul is asking here. This makes it very clear that the unsaved world in Paul’s day understood that Paul was preaching salvation by the grace of God.

Secondly we have that, "all have sinned."

What then? are we better than they? No, in no wise: for we have before proved both Jews and Gentiles, that they are all under sin [Rom. 3:9].

Now Paul doesn’t mean "proved" here. That word is a little too strong; it does not have quite that shade of meaning, because Paul is not trying to prove man a sinner. Rather, he is showing that God judges sin. He assumes man is a sinner, and you don’t have to assume it—it is evident. He is merely stating that which is very obvious today. The better word is charged—“for we have before charged both Jews and Gentiles, that they are all under sin.” He is just stating the case, by the way, that it doesn’t make any difference who we are today—high or low, rich or poor, good or bad—we’re all under sin.

Third is God's righteousness through faith:

But now the righteousness of God without the law is manifested, being witnessed by the law and the prophets [Rom. 3:21].

"The righteousness of God" should be a righteousness of God, since the article is absent in the Greek. This "righteousness" is not an attribute of God—He says that He will not share His glory with another—nor is it the righteousness of man. God has already said that “… our righteousness is as filthy rags …” (Isa. 64:6), and God is not taking in dirty laundry.

And lastly: boasting excluded:

Where is boasting then? It is excluded. By what law? of works? Nay: but by the law of faith [Rom. 3:27].

If God is saving by faith in Christ and not by your merit, your works, then where is boasting? What is it that you and I have to crow about? We can’t even boast of the fact that we’re fundamental in doctrine. We have nothing to glory in today. Paul asks, “Where is boasting then?” And he answers the question he raises.
“It is excluded. By what law? of works? Nay: but by the law of faith.”