Read: Matthew 5:17-20
For I tell you, unless your righteousness exceeds that of the scribes and Pharisees, you will never enter the kingdom of heaven. (Matthew 5:20 ESV)
The scribes and Pharisees have gotten nearly 2000 years worth of bad press from Christianity. They are often painted as the primary antagonists in the Gospels. And while they certainly went to ridiculous lengths to make Jesus' life difficult, at one time they were thought to be the pinnacle of righteousness among the Jews.
During his famous Sermon on the Mount, Jesus told his followers that their righteousness must surpass that of the scribes and Pharisees. That it was the only way in which they would be able to enter God's kingdom. For those of us who are familiar with the Gospel stories that seems like an easy achievement, but for the people who heard Jesus make this pronouncement, it was probably a fairly difficult thing to imagine.
I believe Jesus' purpose for this statement was to call his followers to an authentic relationship with God. A life of love and service to God that surpassed all of the rules and stifling Law that the Pharisees were so fond of in favor of a life-giving Spirit-empowered walk with the Father. A call to embrace the Spirit of the Law, not by adding a bunch of man made statutes to it, but by embracing what it means to show love to one anther.
It also means that we are to be fully devoted followers of Jesus. After all, the only righteousness we have is his. If we are saved by his merciful grace, and covered by his own righteousness then I'd say that is a pretty sure bet for having surpassed that of the scribes and Pharisees.