Answer
The precise identity of the “angel of the Lord” is not given in the Bible. However, there are many important “clues” to his identity. There are Old and New Testament references to “angels of the Lord,” “an angel of the Lord,” and “the angel of the Lord.” It seems when the definite article “the” is used, it is specifying a unique being, separate from the other angels. The angel of the Lord speaks as God, identifies Himself with God, and exercises the responsibilities of God (Genesis 16:7-12; 21:17-18; 22:11-18; Exodus 3:2; Judges 2:1-4; 5:23; 6:11-24; 13:3-22; 2 Samuel 24:16; Zechariah 1:12; 3:1; 12:8). In several of these appearances, those who saw the angel of the Lord feared for their lives because they had “seen the Lord.” Therefore, it is clear that in at least some instances, the angel of the Lord is a theophany, an appearance of God in physical form.
The appearances of the angel of the Lord cease after the incarnation of Christ. Angels are mentioned numerous times in the New Testament, but “the angel of the Lord” is never mentioned in the New Testament after the birth of Christ. One possible difficulty is that the angel who appears to Joseph in a dream in Matthew 1:24 is called "the" angel of the Lord. However, this angel is clearly the same one appearing in verse 20, which calls him "an angel." Matthew is simply referencing the same angel he had just mentioned. There is also some confusion regarding Matthew 28:2, where the KJV says “the angel of the Lord” descended from heaven and rolled the stone away from Jesus’ tomb. It is important to note that the original Greek has no article in front of angel; it could be “the angel” or “an angel,” but the article must be supplied by the translators. Other translations besides the KJV say it was “an angel,” which is the better wording.
It is possible that appearances of the angel of the Lord were manifestations of Jesus before His incarnation. Jesus declared Himself to be existent “before Abraham” (John 8:58), so it is logical that He would be active and manifest in the world. Whatever the case, whether the angel of the Lord was a pre-incarnate appearance of Christ (Christophany) or an appearance of God the Father (theophany), it is highly likely that the phrase “the angel of the Lord” usually identifies a physical appearance of God.
