  We show that $~N=1$~ {\it supersymmetric} Kadomtsev-Petviashvili (SKP)
equations can be embedded into recently formulated $~N=1$~ self-dual {\it
supersymmetric} Yang-Mills theories after appropriate dimensional reduction and
truncation, which yield three-dimensional supersymmetric Chern-Simons theories.
Based on this result, we also give conjectural \hbox{$N=2~$} SKP equations.
Subsequently some exact solutions of these systems including fermionic fields
are given.
