Codimension zero superembeddings

Superembeddings which have bosonic codimension zero are studied in 3,4 and 6 dimensions. The worldvolume multiplets of these branes are off-shell vector multiplets in these dimensions, and their self-interactions include a Born-Infeld term. It is shown how they can be written in terms of standard vector multiplets in flat superspace by working in the static gauge. The action formula is used to determine both Green-Schwarz type actions and superfield actions.