I've never tried fly fishing for carp myself, but there are some devotees. I understand it is pretty big in Europe already.
Carp are not native to the U.S., they were imported from Europe as a food fish, but Americans did not take to eating them (except during the depression when everything was eaten). The European carp is not the same as the new influx of Asian Carp that do all the jumping. In Michigan archers will go into the shallow water flats where they spawn and shoot them (introduced European Carp) with fishing arrows (harpoon points), while they splash about and then discard them on shore. In Europe, that would be considered a waste.