"Owe no one anything except to love one another; for the one loving the other has fulfilled [the] law. For ‘You shall not commit adultery,’ ‘You shall not kill,’ ‘You shall not steal,’ ‘You shall not covet,’ and if [there is] any other commandment in this word, it is summed up in this: ‘You shall love your neighbor as yourself.’ Love does not work evil to one’s neighbor; therefore love is the fulfilment of [the] law" (Romans 13:8-10, author’s own translation).
If Christians are free from the Mosaic law, in what sense do they "fulfill the law"? There is a clear distinction between doing the law, on one hand (Romans 10:5; Galatians 3:10, 12), and fulfilling the law, on the other (Romans 8:4; 13:8, 10; Galatians 5:14). For those who desire to live under the law’s ordinances, consistency demands that they keep the whole law (Galatians 5:3), whereas in Christ the law is fulfilled by the biblical principle of love.
The law of Moses served its designed purpose in bringing adherents to the Lord Jesus Christ (Galatians 3:19-29); thus the church of Christ is what God intended the people of Israel to become (Galatians 6:16; cf. 3:6–4:7). Accordingly, Christians are not under the law but fulfill the law by living out God’s purpose. The apostle Paul stresses love, not as a rule to be obeyed but as an overarching principle characterizing a new way of life (cf. Galatians 5:13–6:10).
What is "the law of Christ" that disciples of Jesus are to fulfill (Galatians 6:2)? Scholarly opinions range from a loosely-defined "law of love" to a continuance of the Mosaic law (filtered through the teachings of Christ). But note the thoroughly antithetical thrust of Galatians, i.e., the true gospel vs. a false one (1:6-12; 2:5, 14), Paul’s former life as a Jew vs. his new life as a Christian (1:13-23), liberty vs. bondage (2:4; 4:1-9; 5:1), circumcision vs. uncircumcision (2:7-9; 5:2-4), works of law vs. faith of Christ (2:16, 20; 3:2, 5-27), and flesh vs. spirit (3:3; 4:21-31; 5:5-26). Within this context it is apparent that the law of Christ is separate and distinct from the law of Moses. From the very beginning of the epistle, the crucial role of the truth of the gospel is highlighted (1:1-12, 23; 2:1-9, 14-21; 3:1-14, 22-29; 4:4-7; 5:7, 13). The law of Christ is simply the practical expression of Christlike love, exemplified in the teachings and example of Christ, and revealed in and obeyed as the gospel of Christ (cf. 1 Corinthians 9:21; Romans 3:27; 8:2).
--Kevin L. Moore
Related Posts: Is the Law of Moses Still Binding?, Was Paul Anti-Law?
In Matthew 5:17-18 Jesus said, "Think not that I came to nullify the law or the prophets; I came not to nullify but to fulfill. For truly I say to you, until the heaven and the earth pass away, one iota or one pen stroke by no means will pass away from the law, until all things come to pass" (author’s own translation).
The Lord spoke these words to a Jewish audience early in his ministry, at least two years before his new covenant was ratified. He assures his listeners that his purpose was not to nullify (destroy, overthrow, abrogate) the law or the prophets. Jesus himself was an Israelite who was amenable to the Jewish law (Galatians 4:4; cf. Matthew 8:4), and he kept it perfectly. Rather, his purpose was to fulfill [plēroō = to fill up or make full] all that the law and the prophets had said concerning the promised Messiah. In fact, one of the primary aims of Matthew’s Gospel is to establish the fact that Jesus and all that was accomplished in his ministry were in fulfillment of the prophetic scriptures (Matthew 1:22; 2:15, 17, 23; 4:14; 8:17; 12:17; 13:35; 21:4; 26:54, 56; 27:9, 35). Accordingly, these things were not to remain unfulfilled for millennia after the Lord’s death, resurrection, and ascension. Once Jesus accomplished his mission on earth, the role that the law and the prophets had served for centuries was complete (see Galatians 3:16-25; Hebrews 8:6-13). In other words, when the intended purpose was fulfilled, that purpose then became obsolete.
Contrary to what many have inferred, the Lord did not say that the Mosaic law was to remain binding until the end of time. Twice in this passage Jesus uses the expression heōs an ("until"). From the standpoint of his contemporary Jewish audience, heaven and earth could pass away at any time. Yet the Lord affirms that until that happens (whenever that might be), nothing will fail from the law or the prophets until all things come to pass. It is a statement of assurance, i.e., the law will unquestionably be vindicated and will have served its purpose when Jesus has completed his personal mission on earth (cf. John 4:34; 5:36; 17:4; 19:30).
--Kevin L. Moore
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Jewish Food Restrictions,
Tithing,
James & Moses' Law,
Was Paul Anti-Law?
"Owe no one anything except to love one another; for the one loving the other has fulfilled [the] law. For ‘You shall not commit adultery,’ ‘You shall not kill,’ ‘You shall not steal,’ ‘You shall not covet,’ and if [there is] any other commandment in this word, it is summed up in this: ‘You shall love your neighbor as yourself.’ Love does not work evil to one’s neighbor; therefore love is the fulfilment of [the] law" (Romans 13:8-10, author’s own translation). 
