Jews affirm that shellfish and other water creatures lacking fins and scales are forbidden. The Torah permits from the waters only what has both fins and scales (Leviticus 11:9-12; Deuteronomy 14:9-10) and calls everything else an abomination. Shrimp, lobster, crab, clams, oysters, and the like therefore fall outside the permitted category, as do fish such as the eel that lack true scales. The rule is one criterion within the wider system of kashrut governing which creatures may be eaten.
The fins-and-scales test has generated genuine rabbinic disagreement at its margins, most famously over sturgeon and swordfish, whose scales, if present at any life stage, differ structurally from the easily removable scales typical of unambiguously kosher fish; nineteenth and twentieth-century authorities split over whether such scales satisfy the halakhic definition, with some Orthodox authorities permitting sturgeon and others prohibiting it, a live disagreement that still produces different kosher certifications for the same species depending on which rabbinic ruling a given supervision agency follows. The fins-and-scales criterion is worked out at length in the Talmud, specifically tractate Chullin, where the sages state as a general principle, at 66b, that any fish possessing scales will also be found to have fins, making scales the practical test a buyer or a rabbinic authority actually applies, with the fins largely assumed rather than separately checked.