Jesus interpreted the scripture concerning the coming of Elijah as being fulfilled by John the Baptist. Jesus said this to his disciples during the descent from the Mount of Transfiguration after Moses and Elijah had appeared with Jesus in glory.

And his disciples asked him, saying, Why then say the scribes that Elias must first come? And Jesus answered and said unto them, Elias truly shall first come, and restore all things. But I say unto you, That Elias is come already, and they knew him not, but have done unto him whatsoever they listed. Likewise shall also the Son of man suffer of them. Then the disciples understood that he spake unto them of John the Baptist. ~ Matthew 17:10-13 (See also Mark 9:11-13)

The scribes taught that Elijah would come before the Messiah and that is why the disciples asked “Why then say the scribes that Elias must first come?” The three disciples had just had an experience in which they had seen both Moses and Elijah with Jesus, and God himself confirmed that Jesus is the Messiah, the Son of God.

In view of all this they asked Jesus, “Why do the scribes say Elijah must come first?” They asked this because things were not adding up. Jesus clears up the confusion when he says, “Elias truly shall first come, and restore all things, and he has already come.” That is why verse 13 says, “Then the disciples understood that he spoke unto them of John the Baptist.”

In the gospel of Luke, the angel Gabriel had said the following to Zechariah concerning John the Baptist before his birth:

“And many of the children of Israel shall he turn to the Lord their God. And he shall go before him in the spirit and power of Elias, to turn the hearts of the fathers to the children, and the disobedient to the wisdom of the just; to make ready a people prepared for the Lord.” ~ Luke 1:16-17 

According to the angel Gabriel, John the Baptist fulfilled the prophecy from Malachi concerning Elijah, who would come and turn the hearts of the fathers to the children. This does not mean that John the Baptist was personally Elijah, but rather, he was like Elijah in the effectiveness of his ministry. 

The the coming of Elijah was fulfilled in a person: John the Baptist. It was not fulfilled in a revival, or a movement. The coming of Elijah is not an “end-time” movement, as some may think.

Jesus did not teach that before his return there would be “days of Elijah.” Jesus interpreted the prophecy about the coming of Elijah as being John the Baptist, and so did the angel Gabriel.

The New Testament emphasizes that we are to look for the coming of Jesus, and not Elijah.