Jews trace their heritage through the male line and always have. Otherwise the tribes would have woman's names rather than mens. The Jewish faith traces it's path through the female line. For example, I I had maried a good Jewish girl all my boys would be thought of as Jews. Since I did not they can still convert to the jewish faith, but they are not Jewish. If they married a good Jewish girl then their children would be consitered Jews. According to halakha, a child is not Jewish if the child's mother is not Jewish.
Reform and Liberal Judaism do not accept the halakhic rules as binding, and accept a child of one Jewish parent, whether father or mother, as Jewish if the parents raise the child as a Jew and the child fosters a Jewish identity, noting that "in the Bible the line always followed the father, including the cases of Joseph and Moses, who married into non-Israelite priestly families.
According to the traditional Rabbinic view, which is maintained by all branches of Orthodox Judaism and Conservative Judaism today, only halakha can define who is or is not a Jew when a question of Jewish identity, lineage, or parentage arises about any person seeking to define themselves or claim that they are Jewish.
As a result, mere belief in the principles of Judaism does not make one a Jew. Similarly, non-adherence by a Jew to the 613 Mitzvot, or even formal conversion to another faith, does not make one lose one's Jewish status. Thus the immediate descendants of all female Jews (even apostates) are still considered to be Jews, as are those of all their female descendants. Even those descendants who are not aware they are Jews, or practice a faith other than Judaism, are defined by this perspective as Jews, as long as they come from an unbroken female line of descent. As a corollary, the children of a Jewish father and a non-Jewish mother are not considered to be Jews by halakha unless they formally convert, even if raised fully observant in the mitzvot.
