I have heard many people (both Christian and non-Christian alike) say that Jesus contradicted the Law of Moses when He instructed His followers not to seek an eye for an eye or a tooth for a tooth. However, a comparison of Christ's instruction in Matthew with the teachings of the Old Testament will reveal that He was only correcting a misunderstanding about the Law and not contradicting the Law itself.
Here is what Jesus said in the Sermon on the Mount about the law of retaliation:
"Ye have heard that it hath been said, An eye for an eye, and a tooth for a tooth: But I say unto you, That ye resist not evil: but whosoever shall smite thee on thy right cheek, turn to him the other also. And if any man will sue thee at the law, and take away thy coat, let him have thy cloke also. And whosoever shall compel thee to go a mile, go with him twain. Give to him that asketh thee, and from him that would borrow of thee turn not thou away."
In this passage and throughout the Sermon on the Mount Jesus was actually clarifying the Law not contradicting the Law. In this particular example, he was explaining that the common understanding of the law requiring an eye for an eye, etc. was incorrect.
Apparently, the Jews of His day understood that law as permission for seeking vengeance against their neighbors, but in reality, that law was given as a guide for judges to follow when sentencing criminals. This is obvious from the context of each occurrence of the phrase in the Old Testament. The clearest such occurrence is found in Deuteronomy 19:18-21 which is introduced by the statement: "And the judges shall make diligent inquisition."
Jesus criticized the common understanding that this law justified vengeance and taught instead that the people should seek to do good to their adversaries and leave vengeance in the hands of God. Christ's instructions fit perfectly with the teachings of the Law and the rest of the Old Testament regarding vengeance. Consider these passages for example:
"Thou shalt not avenge, nor bear any grudge against the children of thy people, but thou shalt love thy neighbour as thyself: I am the LORD." - Leviticus 19:18
"Say not thou, I will recompense evil; but wait on the LORD, and he shall save thee." - Proverbs 20:22
"Say not, I will do so to him as he hath done to me: I will render to the man according to his work." - Proverbs 24:29
"The LORD is good unto them that wait for him, to the soul that seeketh him. It is good that a man should both hope and quietly wait for the salvation of the LORD ... He giveth his cheek to him that smiteth him: he is filled full with reproach. For the Lord will not cast off for ever: But though he cause grief, yet will he have compassion according to the multitude of his mercies." - Lamentations 3:25-32
Had the Jews considered the implications of these passages, they would have realized that the eye for an eye instructions were given to the judges of the land because they were the emissaries of God. They were placed in their positions as judges specifically to carry out God's vengeance (Deuteronomy 1:17, II Chronicles 19:6, Romans 13:4). They should also have realized, however, that the common citizen was not the emissary of God's vengeance. The job of the citizen is to seek peace by doing good to all and letting God take care of punishing those who do wrong.
Thus, Christ's explanation that we should seek good and not vengeance is perfectly in line with the teachings of the Law of Moses. Jesus did not contradict the Law. He merely corrected an erroneous view of the Law by reminding the Jews of other commandments that they were not following.
"Give instruction to a wise man, and he will be yet wiser: teach a just man, and he will increase in learning." (Proverbs 9:9)