  The structure of on-shell and off-shell 2D, (4,4) supersymmetric scalar
multiplets is investigated, in components and in superspace. We reach the
surprising result that there exist eight {\underline {distinct}} on-shell
versions and an even greater variety of off-shell ones. The off-shell
generalised tensor and relaxed N = 4 multiplets are introduced in superspace,
and their universal invariant self-interaction is constructed.
