We　consider　stability　problems　for　the　compressible　viscous　and　diffusive　magnetohydrodynamic　(MHD)　equations　arising　in　the　modeling　of　magnetic　field　confinement　nuclear　fusion.　In　the　first　part,　we　investigate　the　Cauchy　problem　to　the　barotropic　MHD　system.　With　the　help　of　the　techniques　of　anti-symmetric　matrix　and　an　induction　argument　on　the　order　of　the　space　derivatives　of　solutions　in　energy　estimates,　we　prove　that　smooth　solutions　exist　globally　in　time　near　the　non-constant　equilibrium　solutions.　We　also　obtain　the　asymptotic　behavior　of　solutions　when　the　time　goes　to　infinity.　The　result　shows　that　gradients　of　both　the　velocity　and　the　magnetic　field　converge　to　the　equilibrium　solutions　with　the　same　norm　||center　dot||Hs-3,　while　the　density　converge　with　stronger　norm　||center　dot||Hs-1.　In　the　second　part,　the　initial　value　problem　to　the　full　MHD　system　is　studied.　By　means　of　the　techniques　of　choosing　a　non-diagonal　symmetrizer　and　elaborate　energy　estimates,　we　prove　the　existence　and　uniqueness　of　global　solutions　to　the　system　when　the　initial　data　are　close　to　the　non-constant　equilibrium　states.　We　find　that　both　the　density　and　temperature　converge　to　the　equilibrium　states　with　the　same　norm　||center　dot||Hs-1.　These　phenomena　on　the　charge　transport　show　the　essential　relationship　of　the　equations　between　the　barotropic　and　the　full　MHD　systems.