|| Consider a closed surface of genus at least 2. We first recall the non-abelian Hodge theory, which relates the moduli space of Higgs bundles and the moduli space of surface group representations. Next we introduce the Hitchin fibration, which gives a fibration structure of the moduli space of Higgs bundles. As a special family in the moduli space of Higgs bundles, we study cyclic Higgs bundles and show several domination results. Lastly, based on the results we obtained for the cyclic case, we describe a conjectural picture of the whole moduli space.