On a few occasions in the Old Testament period, the angel of the LORD could have been Jesus Himself. The Bible also says the angel of the LORD appeared in New Testament times. What is his identity there?

Matthew

The angel of the LORD appeared to Joseph and foretold the birth of Jesus.

But just when he had resolved to do these things, the angel of the LORD appeared to him in a dream and said, "Joseph, son of David, do not be afraid to take Mary as your wife, for the child conceived in her is from the Holy Spirit. She will bear a son, and you are to name Him Jesus, for He Himself will save His people from their sins" (Matthew 1:20,21).

In this context the angel is not identified.

Identified With Gabriel

Luke, however, identifies the angel of the LORD as Gabriel.

And there appeared unto him [Zechariah] the angel of the LORD standing on the right side of altar of incense (Luke 1:11).

The angel replied, "I am Gabriel. I stand in the presence of God, and I have been sent to speak to you and to bring you this good news (Luke 1:19).

Therefore, in this particular case, the angel of the LORD could not have been Christ.

After The Birth Of Christ

The angel of the LORD also appeared to Joseph after Jesus had been born. He warned Joseph that Herod was about to kill the child.

Now after they had left, behold the angel of the Lord appeared to Joseph in a dream and said, "Get up, take the child and His mother, and escape to Egypt, and remain there until I tell you; for Herod is about to search for the child, to destroy Him" (Matthew 2:13).

He also spoke to Joseph after Herod had died. The angel told him that his family could return from Egypt.

When Herod died, the angel of the LORD suddenly appeared in a dream to Joseph in Egypt and said, "Get up, take the child and His mother, and go to the land of Israel, for those who were seeking the child's life are dead" (Matthew 2:19,20).

Since Jesus had already been born, it is not possible that He could be both the young child, and the angel of the LORD, at the same time. Therefore the angel, or messenger, of the LORD in this context, was someone else.

Delivered Peter

The angel of the LORD helped Peter escape from jail.

Suddenly the angel of the LORD appeared and a light shone in the cell. He touched Peter on the side and woke him, saying, "Get up quickly." And the chains fell off his hands (Acts 12:7).

Killed Herod

The angel of the LORD is the one who put the evil king Herod to death.

And immediately, because he had not given the glory to God, the angel of the LORD struck him down, and he was eaten by worms and died (Acts 12:23).

Though the angel of the LORD is not identified in these instances, there is nothing to suggest that the he was the LORD Himself.

Gabriel

The most likely candidate for the angel of the LORD in the New Testament is Gabriel. When the angel of the LORD appeared to Zechariah to announce the birth of John the Baptist, he identified himself as Gabriel. Though we are not specifically told the identity of this angel of the LORD in the other New Testament contexts, it is consistent to identify him with the angel Gabriel.

Summary

While the angel of the LORD may have been Jesus Christ making temporary appearances on certain occasions in the Old Testament, it certainly was not the same person appearing during the New Testament era. On two different occasions when the angel of the LORD appeared to Joseph, Jesus had already been born. Obviously it could not have been Him. This tells us that the phrase "angel of the LORD" either speaks of more than one person, or that the same person was involved in all the appearances under the title "angel of the LORD." If this is the case, then it could not have been the Lord Jesus in those Old Testament appearances, since it was not He who appeared under this title during the New Testament period. Since Gabriel is identified as the angel of the LORD in Luke, we should probably assume that it is he who appears in other contexts in the New Testament, but is not named.