Who is a Jew?
Jewish identity has both a Halachic definition and a wider self-understanding. The two do not always coincide.

Summary. Halachically, a Jew is someone born to a Jewish mother (matrilineal descent) or someone who has undergone a valid conversion. This standard is the consensus of the Mishnah, Talmud, and Shulchan Aruch and is the operating definition of Orthodox and Conservative Judaism. Reform Judaism in the United States since 1983 also recognizes patrilineal descent when accompanied by Jewish upbringing. Beyond Halacha, Jewishness encompasses ethnicity, culture, history, language, and peoplehood. A Jew is also someone who shows up.
The Halachic rule of matrilineal descent is sourced in Devarim 7:3–4 and elaborated in Talmud Kiddushin 68b, where the Sages derive from the verse that a child’s status follows the mother. The Mishneh Torah (Hilchot Issurei Biah 15:3) and the Shulchan Aruch (Even Ha’Ezer 4:19) codify this as binding.
Conversion (giyur) is the second route. A valid Orthodox conversion requires acceptance of the mitzvot (kabbalat ol mitzvot), circumcision (for men), and immersion in a mikveh (ritual bath), all under the supervision of a beit din (rabbinical court) of three observant Jews. Conservative conversions are conducted on similar lines with some halachic variation; Reform conversions typically omit the requirement of immersion and circumcision in some communities.
But Jewishness is wider than the strictly Halachic definition. A Jew is a member of the Jewish people, the descendants of Avraham, Yitzchak, and Yaakov, called Yisrael — “the one who wrestles with God” (Bereshit 32:29). The Jewish people are at once an ethnos, a faith, a people, a culture, and a project — none of which fully captures the whole. A Jew is also someone who, in moments of crisis or commitment, finds himself or herself irreducibly Jewish.
The State of Israel’s Law of Return (1950, amended 1970) is deliberately broad, but its breadth lies in eligibility rather than in the definition of a Jew: the law itself defines a Jew as someone born to a Jewish mother or who has converted and is not a member of another religion, while the 1970 amendment extends the right of return to the child or grandchild of a Jew, and to their spouses. This is not a Halachic definition; it is a definition of who would have been persecuted as a Jew, and therefore who deserves refuge under Israel’s founding promise.
Where Denominations Diverge
Orthodox: matrilineal descent or Orthodox conversion only. Conservative: matrilineal descent or Conservative/Orthodox conversion. Reform (US, since 1983): matrilineal OR patrilineal descent with Jewish upbringing; conversion under Reform auspices. Reconstructionist: patrilineal descent recognized since 1968. Renewal: similar to Reconstructionist. Reform in the UK and Israel typically retains matrilineal descent only.
Sources
Torah: Devarim 7:3–4; Bereshit 32:29.
Talmud Bavli, Kiddushin 68b.
Rambam, Mishneh Torah, Hilchot Issurei Biah 15:3.
Shulchan Aruch, Even Ha’Ezer 4:19.
CCAR Responsum on Patrilineal Descent (1983).
Israel Law of Return (1950, 1970 amendment).
Israel's Basic Laws: An Introduction to the Law of Return — Jewish Virtual Library.
Judaism: Patrilineal Descent — Jewish Virtual Library.
Why Is Jewishness Matrilineal? Maternal Descent In Judaism — Chabad.org.
Yevamot 47b — Sefaria.
Further Reading
Shaye J.D. Cohen, The Beginnings of Jewishness.
Hayim Halevy Donin, To Be a Jew.
Anita Diamant, Choosing a Jewish Life.
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