Who are the Messianic Jews? What is the Messianic Jewish movement?
For most of the leaders in the movement, these two questions are tied together.
The Messianic Jews are those Jews who believe in Jesus of Nazareth as Messiah of Israel, as Lord and Saviour (or Yeshua as they would say) and gather themselves into congregations of Jewish believers in Yeshua so as to live a corporate life as Jewish disciples of their Messiah.
There are people who call any Jew who believes in Jesus a Messianic Jew. In my opinion, this is not a helpful way of speaking and can cause confusion.
Jewish and Christian faith is inherently a shared faith in covenantal relationships. To call any person of Jewish roots who believes in Jesus a Messianic Jew can also lead to vastly inflated and unrealistic figures of how many Messianic Jews there are.