  The formalism of integrable mappings is applied to the problem of
constructing hierarchies of $(1+2)$ dimensional integrable systems in the
$(2|2)$ superspace. We find new supersymmetric integrable mappings and
corresponding to them new hierarchies of integrable systems which, at the
reduction to the $(1|2)$ superspace, possess $N=2$ supersymmetry. The general
formulae obtained for the hierarchies are used to explicitly derive their first
nontrivial equations possessing a manifest $N=2$ supersymmetry. New bosonic
substitutions and hierarchies are obtained from the supersymmetric counterparts
in the bosonic limit.
