What does Romans 6:1 mean?

Paul begins this chapter by posing a question about the implications of the statements that ended chapter 5. There, he wrote that where sin increased, God's grace "super-increased." That is, as sin increased, so did God's grace abound to cover the sin of all those who trusted in Christ's death to cover their sin. We literally cannot out-sin the grace of God.

What does that mean, though, for those who have been reconciled to God through faith in Christ? What are Christians supposed to do about sin now that we are Christians? As Paul asks here, should we just keep sinning so that God's grace can just keep increasing? This seems to have been a common criticism of Paul's teaching, as it is one he refutes often in his writings (Romans 3:8; 2 Corinthians 5:17; Galatians 5:19–24). It's a frequent charge against Christianity, even today, suggesting that the gospel is really just a license to sin. In the following verse, Paul will answer this slanderous charge with an emphatic "no!"