Matthew 5:17-20 (KJV) 
 17  Think not that I am come to destroy the law, or the prophets: I am not come to destroy, but to fulfil. 
 18  For verily I say unto you, Till heaven and earth pass, one jot or one tittle shall in no wise pass from the law, till all be fulfilled. 
 19  Whosoever therefore shall break one of these least commandments, and shall teach men so, he shall be called the least in the kingdom of heaven: but whosoever shall do and teach them, the same shall be called great in the kingdom of heaven. 
 20  For I say unto you, That except your righteousness shall exceed the righteousness of the scribes and Pharisees, ye shall in no case enter into the kingdom of heaven. 

Jesus said that he came not to destroy the law, but to fulfill it.  What does it mean to fulfill the law?  The word fulfill also means to complete with a purpose or design or to bring about its full meaning. All of these meanings are appropriate here.  Jesus fulfilled the law in that he not only lived out the law perfectly without sin, but that he himself was a perfect sacrifice required by the law to be a substitution for sin.  The law is not limited to only the moral code that was given to Moses as listed in the Ten Commandments, but is used of the entire Jewish system of worship including the temple system or worship and all the food and purity laws.  So Jesus’ fulfilling of the law extends beyond living a perfect moral life.  The entire structure of worship that was established by God with the Jews was done as a foreshadow of what was to come in the form of Christ.

But does verse 18 mean that we are still under the law?  The law is still in effect in that it continues to bring the sinner to repentance by making them aware of their sin.  Without the law man would not know what the moral will of God was.  Paul writes in Romans 7:7,

⁷What shall we say then? Is the law sin? God forbid. Nay, I had not known sin, but by the law: for I had not known lust, except the law had said, Thou shalt not covet.

The law was the greatest expression of God’s morality until Christ.  So the law continues to be the law, but Christ has become an even better expression of God’s morality than the written law.   

What Paul taught us is so important to understand here: that if we try to keep the law, we are denying that Jesus fulfilled the law.  If we live as though we were under the law, we are no longer under the grace of God. 

The role of the law has caused a lot of confusion in Paul’s day and continues to today.  There is nothing wrong with the law and Jesus did not need to destroy it.  However, what Jesus did was to show us what it looks like when someone keeps the law perfectly.  Jesus expression of the law is the highest form of righteousness than could be expressed to mankind in that way Jesus is greater than the law (Matt 12:6-8).  So the law is still in effect, but it is now much better expressed through Christ.  We keep the law not by living it through our will to be righteous, but by the submission of our will to Jesus. If you truly want to be righteous than live by these sayings of Jesus in his Sermon and you will be obeying the law. 

Jesus told us that we must exceed the righteousness of the Pharisees who were very religious and struggled to keep the law perfectly.  They fasted and prayed and offered sacrifices frequently as well as tenaciously kept the Sabbath.  How many of us could really be good Pharisees today?  The point that Jesus was making was that our righteousness exceeds that of the Pharisees not in that we are trying harder, but rather that we are loving the law of God more accurately as we love it as expressed by Christ.

If you still want to live by the law, then obey the Sermon on the Mount for by so doing you will live the law more perfectly.  There are some today that think they must keep the food laws given to Moses in order to keep their bodies clean. Paul taught very clearly here in Romans 7:1-6 that we died to the law through Christ (Baptism).

4 Wherefore, my brethren, ye also are become dead to the law by the body of Christ; that ye should be married to another, even to him who is raised from the dead, that we should bring forth fruit unto God.  

So we are now free from the law through our baptism into Christ. We did not earn salvation through our works or goodness.  It was gift given to us through the suffering and sacrifice of Jesus that we might die with him and receive the gift of eternal life. Let us be eternally grateful to him for our freedom rather than become bound to the law.