What is the Messiah supposed to accomplish? One of the central themes of biblical prophecy is the promise of a future age of perfection characterized by universal peace and recognition of God. (Isaiah 2:1-4, 32:15-18, 60:15-18; Zephaniah 3:9; Hosea 2:20-22; Amos 9:13-15; Micah 4:1-4; Zechariah 8:23, 14:9; Jeremiah 31:33-34)
Specifically, the Bible says he will:
Build the Third Temple (Ezekiel 37:26-28).
Gather all Jews back to the Land of Israel (Isaiah 43:5-6).
Usher in an era of world peace, and end all hatred, oppression, suffering and disease. As it says: "Nation shall not lift up sword against nation, neither shall man learn war anymore." (Isaiah 2:4)
Spread universal knowledge of the God of Israel, which will unite humanity as one. As it says: "God will be King over all the world ― on that day, God will be One and His Name will be One" (Zechariah 14:9).
If an individual fails to fulfill even one of these conditions, then he cannot be the Messiah.
Because no one has ever fulfilled the Bible's description of this future King, Jews still await the coming of the Messiah. All past Messianic claimants, including Jesus of Nazareth, Bar Cochba and Shabbtai Tzvi have been rejected.
Christians counter that Jesus will fulfill these in the Second Coming. Jewish sources show that the Messiah will fulfill the prophecies outright; in the Bible no concept of a second coming exists.