  The paper is devoted to the study of compositions of polyharmonic mappings in
simply connected domains. More precisely, we determine necessary and sufficient
conditions of polyharmonic mapping $f$ such that $f\circ F$ (resp. $F\circ f$)
is $l$-harmonic for any analytic function (or harmonic mapping but not
analytic, or $q$-harmonic mapping but not $(q-1)$-harmonic) $F$.
