“Do not think that I came to destroy the Law or the Prophets. I did not come to destroy but to fulfill. For assuredly, I say to you, till heaven and earth pass away, one jot or one tittle will by no means pass from the law till all is fulfilled. Whoever therefore breaks one of the least of these commandments, and teaches men so, shall be called least in the kingdom of heaven; but whoever does and teaches them, he shall be called great in the kingdom of heaven.”
Listen to the "Verse by Verse" episode covering this scripture.
Some misunderstand Jesus Christ’s mission, feeling He did away with the teachings of the Old Testament and replaced them with new teachings. But Christ made clear that His teachings were a continuation of the teachings He inspired to be a part of the Holy Scriptures.
It was the One who became Jesus Christ who thundered the 10 Commandments from Mount Sinai and spoke to Moses (1 Corinthians 10:4; see “Jesus in the Old Testament?”).
What does it mean to “fulfill” the law? The Greek word translated “fulfill” is pleroo, which means “to make full, to fill to the full” (Vine’s Complete Expository Dictionary of Old and New Testament Words).
Jesus Christ continued in the Sermon on the Mount (Matthew 5, 6 and 7) to expand the meaning of God’s law, to show its spiritual intent. By doing this, He gave fuller understanding and magnified and exalted the law (Isaiah 42:21). He in no way destroyed or annulled it.
In Matthew 5:19 Jesus clearly tells His followers to obey even what someone might consider “one of the least of these commandments.”
The essence of the New Covenant is not a change in the commandments, but the fact that they can be written on our hearts and minds by the Holy Spirit dwelling in Christians (Hebrews 10:16). This allows us to internalize and fully obey God in a way ancient Israel never could.