Monkfish are not kosher, because they do not have scales. In order for a fish to be classed as kosher it must have both fins and scales (see Leviticus 11:9).

It is interesting to note that many monkfish are bottom-feeders. Nachmanides points out that many non-kosher fish live deeper in the murky depths, while kosher species are more likely to be found closer to the surface, where the water is clearer and healthier. There are, however, exceptions, and the place where a fish typically feeds is not reason to class it as kosher or unkosher.

For more, see All About Kosher Fish.