In Hebrews 8, the author finally introduces the term covenant, the covenant ensured by the sacrifice of the Lord Jesus. Notice that what he says: "But the ministry Jesus has received is as superior to theirs as the covenent of which he is mediator is superior to the old one" (Hebrews 8:13 NIV,84). This means that the covenant is better. It can be seen as discontinuous with the previous covenant and therefore understood to be completely new. Others believe that it is a continuation, a fulfillment of the covenant of the Old Testament. The author makes a reference later in Chapter 8 to the covenant the Prophet Jeremiah mentions in chapter 33. He says that there will be a time when the Lord establishes a new covenant. Let me make clear that, for Jeremiah, the new covenant would take place in the future. And here we are looking at a contrast; or better, it is either a continuation or something completely new. There's the dilemma. Now Christians are divided on this issue.

My personal opinion is that the new covenant is a continuation; I see the Lord always working throughout history with his people, the for his Jews, and later on, Gentiles, Jews and all humanity. So salvation is by grace. The difference was that in the Old Testament, Jesus had not offered his sacrifice yet, so the people of the Old Testament could not look back to it like we can. We have a better covenant now because salvation is already accomplished, and we should not be anxious about failing, because Jesus has forgiven all our sins. Therefore, the covenant is better but it is also new in the sense that now there are no barriers or limitations imposed by the Law. We don't need to offer same sacrifices, we don't need to have the same laws about food, and we don't have to have the same festivals, etc. Now everything is through faith, trusting in Jesus. So, at the end of chapter 8, the author says, "He has made the first [covenant] obsolete; and what is obsolete and aging will soon disappear" (Hebrews 8:13 NIV84). Therefore, the old covenant is over and the new covenant is its continuation.