This　paper　considers　the　consensus　problems　for　complex　dynamical　network　systems.　For　the　dynamical　network　systems,　more　than　one　interconnections　are　considered　in　view　of　the　fact　that　the　subsystems　of　networks　may　be　multiple　interconnected.　Sufficient　and　necessary　conditions　for　consensus　are　given　for　the　linear　dynamical　networks　with　state　interconnections　and　controller　interconnections.　Moreover,　a　static　output　controller　design　method　based　on　a　parameter-dependent　Lyapunov　function　is　provided.　Furthermore,　multiple　interconnected　nonlinear　dynamical　network　system　is　investigated,　and　the　subsystems　are　Lure　systems.　Kalman-Yakubovich-Popov　(KYP)　lemma　and　the　Schur　complement　formula　are　applied　to　get　novel　criteria,　which　have　the　forms　of　linear　matrix　inequalities　(LMIs).　And　the　test　of　the　consensus　of　a　network　with　Lure　systems　as　its　nodes　is　separated　into　the　test　of　the　absolute　stability　of　several　independent　Lure　systems.　Finally,　numerical　examples　are　presented　to　illustrate　the　efficiency　and　applicability　of　the　proposed　methods.　©　2015　IEEE.