  Firstly, we introduce a class of new algebraic systems which generalize Hopf
quasigroups and Hopf $\pi-$algebras called $Q$-graded Hopf quasigroups, and
research some properties of them. Secondly, we define the representations of
$Q$-graded Hopf quasigroups, i.e $Q$-graded Hopf quasimodules, research the
construction method and fundamental theorem of them. Thirdly, we research the
smash products of $Q$-graded Hopf quasigroups.
