I'm going to give you a complicated answer to what might seem, at first glance, to be a simple question.
Movement of Messianic Judaism
If, by "Messianic Judaism" the current movement including "Jews for Jesus", Hebrew Christians" and the like, the answer is "No". Every branch of Judaism, from Orthodox to Reform, agrees that Jesus was not the Messiah and that one cannot be part of a faith that rejects the belief in Jesus as the Messiah and affirm that he is. Moreover, Judaism rejects the entire concept of virgin birth.
Jewish Yearning for the Messiah
If, as a larger issue, you asked whether Judaism believes in a Messiah, the answer would be a complicated "Yes". The traditional prayerbook includes prayers about the (future) coming of the Messiah, a descendent of David, who will restore the traditional kingdom, rebuild the Temple and usher in a ideal -- albeit imperfect -- age. Mordecai Kaplan, the founder of the Reconstructionist Movement in Judaism, preferred to speak of a "Messianic Era" in which people will act in a humane manner and need not rely upon a supernatural figure.
This is the title of a Christian minister who serves a Messianic congregation. Messianic Judaism is a form of Christianity where the congregants self-identify with Jews or the Jewish people, though most are not actually Jews by birth.
No.
Messianic Judaism and Christianity.
Messianic Judaism and Christianity both believe in Jesus as the Messiah, but they differ in their views on Jewish traditions and practices. Messianic Judaism combines Jewish customs with belief in Jesus, while Christianity typically does not emphasize Jewish traditions. Additionally, Messianic Judaism often observes Jewish holidays and dietary laws, while Christianity generally does not.
Since Messianic Judaism is a form of Christianity, Messianic Jews raise no objections to the mainstream Christian arguments that all foods are permissible according to the New Testament. Some Messianic Jews eat kosher for personal reasons, but the majority of the movement is not so inclined.
No. Messianics, as a whole, are not regonised as Jews by the State of Israel. This includes those who were born to Jewish families but converted to Messianic Christianity. The Supreme Court of Israel has clarified that Messianic Judaism is a form of Christianity and as such Messianic Jews do not qualify for the Right of Return.
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Messianic Judaism is a modern religious movement, and it is difficult to attribute its creation to a single individual. However, it emerged in the mid-20th century as Jewish believers in Jesus sought to reconcile their faith in Jesus as the Messiah with their Jewish identity and heritage. Various leaders and organizations have contributed to the development and spread of Messianic Judaism.
According to Judaism, those Jews who believe in God and do his Mitzvot or Commandments are those who are faithful and will have a share in the World to Come.However, Jews typically do not speak of a faithful remnant; this tends to be a Christian designation for ethnic Israelites/Jews who believe that Jesus Christ is the Messiah. Under this rubric, Messianic Jews* who convert to Messianic Judaism from Traditional Rabbinic Judaism or are their descendants are the "faithful remnant". Such a concept of a "faithful remnant of Judaism" is alluded to in Romans 9:27. Jews would not argue that people who believe in Jesus Christ as the Messiah are faithful Jews, but rather that they have abandoned their faith for Christianity.*See this Related Question which discusses how Messianic Judaism is a form of Protestant Christianity.
I believe in Messianic Judaism as the closest thing to that which God intends for humanity.
No. The Cross is a Christian symbol. It has no value in Judaism.
The main difference between Messianic Jews and other Jews is that Messianic Jews is a merger between evangelical Christianity with elements of Judaism. It is a new religion, developed in the 1960's.