  With the motivation to develop superconformal field theory on S^3, we
introduce a 2n-extended supersphere S^{3|4n}, with n=1,2,..., as a homogeneous
space of the three-dimensional Euclidean superconformal group OSp(2n|2,2) such
that its bosonic body is S^3. Supertwistor and bi-supertwistor realizations of
S^{3|4n} are derived. We study in detail the n=1 case, which is unique in the
sense that the R-symmetry subgroup SO^*(2n) of the superconformal group is
compact only for n=1. In particular, we show that the OSp(2|2,2)
transformations preserve the chiral subspace of S^{3|4}. Several supercoset
realizations of S^{3|4n} are presented. Harmonic/projective extensions of the
supersphere by auxiliary bosonic fibre directions are sketched.
